R S AGGARWAL AND V AGGARWAL Solutions for Class 9 Maths Chapter 10 - Quadrilaterals

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Chapter 10 - Quadrilaterals Exercise MCQ

Question 1

Three angles of quadrilateral are 80°, 95° and 112°. Its fourth angle is

(a) 78°

(b) 73°

(c) 85°

(d) 100°

 

Solution 1

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 2

The angles of a quadrilateral are in the ratio 3:4:5:6. The smallest of these angles is

(a) 45°

(b) 60°

(c) 36°

(d) 48°

 

Solution 2

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 3

In the given figure, ABCD is a parallelogram in which BAD = 75° and CBD = 60°. Then, BDC =?

(a) 60°

(b) 75°

(c) 45°

(d) 50°

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Solution 3

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 4

ABCD is a rhombus such that ACB = 50°. Then, ADB = ?

(a) 40° 

(b) 25° 

(c) 65°  

(d) 130° 

Solution 4

Correct option: (a)

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

ABCD is a rhombus.

AD BC and AC is the transversal.

DAC = ACB  (alternate angles)

DAC = 50° 

In ΔAOD, by angle sum property,

AOD + DAO + ADO = 180° 

90° + 50° + ADO = 180° 

ADO = 40° 

ADB = 40° 

Question 5

In which of the following figures are the diagonals equal?

  1. Parallelogram
  2. Rhombus
  3. Trapezium
  4. Rectangle

 

Solution 5

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 6

If the diagonals of a quadrilateral bisect each other at right angles, then the figure is a

a. trapezium

b. parallelogram

c. rectangle

d. rhombus

Solution 6

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 7

The lengths of the diagonals of a rhombus are 16 cm and 12 cm. The length of each side of the rhombus is

  1. 10 cm
  2. 12 cm
  3. 9cm
  4. 8cm
Solution 7

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

Question 8

The length of each side of a rhombus is 10 cm and one of its diagonals is of length 16 cm. The length of the other diagonal is

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Solution 8

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 9

A diagonal of a rectangle is inclined to one side of the rectangle at 35°. The acute angle between the diagonals is

(a) 55° 

(b) 70° 

(c) 45° 

(d) 50° 

Solution 9

Correct option: (b)

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

DAO + OAB = DAB

DAO + 35° = 90° 

DAO = 55° 

ABCD is a rectangle and diagonals of a rectangle are equal and bisect each other.

OA = OD

ODA = DAO (angles opposte to equal sides are equal)

ODA = 55° 

In DODA, by angle sum property,

ODA + DAO + AOD = 180° 

55° + 55° + AOD = 180° 

AOD = 70° 

Question 10

If ABCD is a parallelogram with two adjacent angles A = B, then the parallelogram is a

  1. rhombus
  2. trapezium
  3. rectangle
  4. none of these

 

Solution 10

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 11

In a quadrilateral ABCD, if AO and BO are the bisectors of A and B respectively, C = 70° and D = 30°. Then, AOB =?

  1. 40°
  2. 50°
  3. 80°
  4. 100°
Solution 11

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 12

The bisectors of any adjacent angles of a parallelogram intersect at

  1. 30°
  2. 45°
  3. 60°
  4. 90°

 

Solution 12

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 13

The bisectors of the angles of a parallelogram enclose a

  1. rhombus
  2. square
  3. rectangle
  4. parallelogram
Solution 13

Correct option: (c)

The bisectors of the angles of a parallelogram enclose a rectangle.

Question 14

If bisectors of A and B of a quadrilateral ABCD intersect each other at P, of B and C at Q, of C and D at R and of D and A at S then PQRS is a

(a) rectangle

(b) parallelogram

(c) rhombus

(d) quadrilateral whose opposite angles are supplementary

Solution 14

Correct option: (d)

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

In ΔAPB, by angle sum property,

APB + PAB + PBA = 180° 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

In ΔCRD, by angle sum property,

CRD + RDC + RCD = 180° 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Now, SPQ + SRQ = APB + CRD

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

= 360° - 180° 

= 180° 

Now, PSR + PQR = 360° - (SPQ + SRQ)

= 360° - 180° 

= 180° 

Hence, PQRS is a quadrilateral whose opposite angles are supplementary. 

Question 15

The figure formed by joining the mid-points of the adjacent sides of a quadrilateral is a

  1. rhombus
  2. square
  3. rectangle
  4. parallelogram
Solution 15

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

Question 16

The figure formed by joining the mid-points of the adjacent sides of a square is a

  1. rhombus
  2. square
  3. rectangle
  4. parallelogram
Solution 16

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 17

The figure formed by joining the mid-points of the adjacent sides of a parallelogram is a

  1. rhombus
  2. square
  3. rectangle
  4. parallelogram
Solution 17

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 18

The figure formed by joining the mid-points of the adjacent sides of a rectangle is a

  1. rhombus
  2. square
  3. rectangle
  4. parallelogram
Solution 18

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 19

The figure formed by joining the mid-points of the adjacent sides of a rhombus is a

  1. rhombus
  2. square
  3. rectangle
  4. parallelogram
Solution 19

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 20

The quadrilateral formed by joining the midpoints of the sides of a quadrilateral ABCD, taken in order, is a rectangle, if

(a) ABCD is a parallelogram

(b) ABCD is a rectangle

(c) diagonals of ABCD are equal

(d) diagonals of ABCD are perpendicular to each other

Solution 20

Correct option: (d)

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

In ΔABC, P and Q are the mid-points of sides AB and BC respectively.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

In ΔADC, R and S are the mid-points of sides CD and AD respectively.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

From (i) and (ii),

PQ RS and PQ = RS

Thus, in quadrilateral PQRS, a pair of opposite sides are equal are parallel.

So, PQRS is a parallelogram.

Let the diagonals AC and BD intersect at O.

Now, in ΔABD, P and S are the mid-points of sides AB and AD respectively.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Thus, in quadrilateral PMON, PM NO and PN MO.

PMON is a parallelogram.

MPN = MON (opposite angles of a parallelogram are equal)

MPN = BOA (since BOA = MON)

MPN = 90° (since AC BD, BOA = 90°)

QPS = 90° 

Thus, PQRS is a parallelogram whose one angle, i.e. QPS = 90°.

Hence, PQRS is a rectangle if AC BD. 

Question 21

The quadrilateral formed by joining the midpoints of the sides of a quadrilateral ABCD, taken in order, is a rhombus, if

(a) ABCD is a parallelogram

(b) ABCD is a rhombus

(c) diagonals of ABCD are equal

(d) diagonals of ABCD are perpendicular to each other

Solution 21

Correct option: (c)

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

In ΔABC, P and Q are the mid-points of sides AB and BC respectively.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

In ΔBCD, Q and R are the mid-points of sides BC and CD respectively.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

In ΔADC, S and R are the mid-points of sides AD and CD respectively.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

In ΔABD, P and S are the mid-points of sides AB and AD respectively.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

PQ RS and QR SP [From (i), (ii), (iii) and (iv)]

Thus, PQRS is a parallelogram.

Now, AC = BD (given)

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

PQ = QR = RS = SP [From (i), (ii), (iii) and (iv)]

Hence, PQRS is a rhombus if diagonals of ABCD are equal. 

Question 22

The figure formed by joining the midpoints of the sides of a quadrilateral ABCD, taken in order, is a square, only if

(a) ABCD is a rhombus

(b) diagonals of ABCD are equal

(c) diagonals of ABCD are perpendicular

(d) diagonals of ABCD are equal and perpendicular

Solution 22

Correct option: (d)

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

In ΔABC, P and Q are the mid-points of sides AB and BC respectively.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

In ΔBCD, Q and R are the mid-points of sides BC and CD respectively.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

In ΔADC, S and R are the mid-points of sides AD and CD respectively.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

In ΔABD, P and S are the mid-points of sides AB and AD respectively.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

PQ || RS and QR || SP [From (i), (ii), (iii) and (iv)]

Thus, PQRS is a parallelogram.

Now, AC = BD (given)

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

PQ = QR = RS = SP [From (i), (ii), (iii) and (iv)]

 

Let the diagonals AC and BD intersect at O.

Now,

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Thus, in quadrilateral PMON, PM || NO and PN || MO.

PMON is a parallelogram.

MPN = MON (opposite angles of a parallelogram are equal)

MPN = BOA (since BOA = MON)

MPN = 90° (since AC BD, BOA = 90°)

QPS = 90° 

Thus, PQRS is a parallelogram such that PQ = QR = RS = SP and QPS = 90°.

Hence, PQRS is a square if diagonals of ABCD are equal and perpendicular. 

Question 23

If an angle of a parallelogram is two-third of its adjacent angle, the smallest angle of the parallelogram is

  1. 108°
  2. 54°
  3. 72°
  4. 81°
Solution 23

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 24

If one angle of a parallelogram is 24° less than twice the smallest angle, then the largest angles of the parallelogram is

  1. 68°
  2. 102°
  3. 112°
  4. 136°

 

Solution 24

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 25

If A, B, C and D of a quadrilateral ABCD taken in order, are in the ratio 3:7:6:4, then ABCD is a

  1. rhombus
  2. kite
  3. trapezium
  4. parallelogram
Solution 25

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 26

Which of the following is not true for a parallelogram?

  1. Opposite sides are equal.
  2. Opposite angles are equal.
  3. Opposite angles are bisected by the diagonals.
  4. Diagonals bisect each other.
Solution 26

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 27

If APB and CQD are two parallel lines, then the bisectors of APQ, BPQ, CQP and PQD enclose a

  1. square
  2. rhombus
  3. rectangle
  4. kite

 

Solution 27

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 28

In the given figure, ABCD is a parallelogram in which BDC = 45° and BAD = 75°. Then, CBD =?

  1. 45°
  2. 55°
  3. 60°
  4. 75°

 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

 

Solution 28

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 29

If area of a gm with side ɑ and b is A and that of a rectangle with side ɑ and b is B, then

(a) A > B

(b) A = B

(c) A < B

(d) A  B

Solution 29

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 30

In the given figure, ABCD is a gm and E is the mid-point at BC, Also, DE and AB when produced meet at F. Then,

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Solution 30

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 31

P is any point on the side BC of a ΔABC. P is joined to A. If D and E are the midpoints of the sides AB and AC respectively and M and N are the midpoints of BP and CP respectively then quadrilateral DENM is

(a) a trapezium

(b) a parallelogram

(c) a rectangle

(d) a rhombus

Solution 31

Correct option: (b)

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

In ΔABC, D and E are the mid-points of sides AB and AC respectively.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Hence, DENM is a parallelogram.

Question 32

The parallel sides of a trapezium are ɑ and b respectively. The line joining the mid-points of its non-parallel sides will be

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Solution 32

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

Question 33

In a trapezium ABCD, if E and F be the mid-points of the diagonals AC and BD respectively. Then, EF =?

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

 

Solution 33

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 34

In the given figure, ABCD is a parallelogram, M is the mid-point of BD and BD bisects B as well as D. Then, AMB=?

  1. 45°
  2. 60°
  3. 90°
  4. 30°

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Solution 34

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 35

In the given figures, ABCD is a rhombus. Then

(a) AC2 + BD2 = AB2

(b) AC2 + BD2 = 2AB2

(c) AC2 + BD2 = 4AB2

(d) 2(AC2 + BD2)=3AB2

 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

 

Solution 35

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 36

In a trapezium ABCD, if AB CD, then (AC2 + BD2) =?

(a) BC2 + AD2 + 2BC. AD

(b) AB2 +CD2 + 2AB.CD

(c) AB2 + CD2 + 2AD. BC

(d) BC2 + AD2 + 2AB.CD

 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Solution 36

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 37

Two parallelogram stand on equal bases and between the same parallels. The ratio of their area is

  1. 1:2
  2. 2:1
  3. 1:3
  4. 1:1
Solution 37

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 38

In the given figure, AD is a median of ΔABC and E is the mid-point of AD. If BE is joined and produced to meet AC in F, then AF =?

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Solution 38

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 39

The diagonals AC and BD of a parallelogram ABCD intersect each other at the point O such that DAC = 30°and AOB = 70°. Then, DBC =?

  1. 40°
  2. 35°
  3. 45°
  4. 50°

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Solution 39

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 40

Three statement are given below:

  1. In a gm, the angle bisectors of two adjacent angles enclose a right angle.
  2. The angle bisectors of a gm form a rectangle.
  3. The triangle formed by joining the mid-point of the sides of an isosceles triangle is not necessarily an isosceles.

Which is true?

  1. I only
  2. II only
  3. I and II
  4. II and III
Solution 40

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 41

Three statements are given below:

 I. In a rectangle ABCD, the diagonal AC bisects A as well as C.

 II. In a square ABCD, the diagonal AC bisects A as well as C

 III. In a rhombus ABCD, the diagonal AC bisects A as well as C

Which is true?

  1. I only
  2. II and III
  3. I and III
  4. I and II
Solution 41

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 42

In a quadrilateral PQRS, opposite angles are equal. If SR = 2 cm and PR = 5 cm then determine PQ. 

Solution 42

Since opposite angles of quadrilateral are equal, PQRS is a parallelogram.

PQ = SR (opposite sides of parallelogram are equal)

PQ = 2 cm

Question 43

Diagonals of a parallelogram are perpendicular to each other. Is this statement true? Give reasons for your answer.

Solution 43

The given statement is false.

Diagonals of a parallelogram bisect each other. 

Question 44

What special name can be given to a quadrilateral PQRS if P + S = 118°?

Solution 44

In quadrilateral PQRS, P and S are adjacent angles.

Since the sum of adjacent angles 180°, PQRS is not a parallelogram.

Hence, PQRS is a trapezium. 

Question 45

All the angles of a quadrilateral can be acute. Is this statement true? Give reasons for your answer.

Solution 45

The given statement is false.

We know that the sum of all the four angles of a quadrilateral is 360°.

If all the angles of a quadrilateral are acute, the sum will be less than 360°. 

Question 46

All the angles of a quadrilateral can be right angles. Is this statement true? Give reasons for your answer.

Solution 46

The given statement is true.

We know that the sum of all the four angles of a quadrilateral is 360°.

If all the angles of a quadrilateral are right angles,

Sum of all angles of a quadrilateral = 4 × 90° = 360° 

Question 47

All the angles of a quadrilateral can be obtuse. Is this statement true? Give reasons for your answer.

Solution 47

The given statement is false.

We know that the sum of all the four angles of a quadrilateral is 360°.

If all the angles of a quadrilateral are obtuse, the sum will be more than 360°. 

Question 48

Can we form a quadrilateral whose angles are 70°, 115°, 60° and 120°? Give reasons for your answer.

Solution 48

We know that the sum of all the four angles of a quadrilateral is 360°.

Here,

70° + 115° + 60° + 120° = 365° 360° 

Hence, we cannot form a quadrilateral with given angles. 

Question 49

What special name can be given to a quadrilateral whose all angles are equal?

Solution 49

A quadrilateral whose all angles are equal is a rectangle. 

Question 50

If D and E are respectively the midpoints of the sides AB and BC of ΔABC in which AB = 7.2 cm, BC = 9.8 cm and AC = 3.6 cm then determine the length of DE.

Solution 50

D and E are respectively the midpoints of the sides AB and BC of ΔABC.

Thus, by mid-point theorem, we have

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

Question 51

In a quadrilateral PQRS, the diagonals PR and QS bisect each other. If Q = 56°, determine R.

Solution 51

Since the diagonals PR and QS of quadrilateral PQRS bisect each, PQRS is a parallelogram.

Now, adjacent angles of parallelogram are supplementary.

Q + R = 180° 

56° + R = 180° 

R = 124° 

Question 52

In the adjoining figure, BDEF and AFDE are parallelograms. Is AF = FB? Why or why not?

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

Solution 52

AFDE is a parallelogram

AF = ED …(i)

BDEF is a parallelogram.

FB = ED …(ii)

From (i) and (ii),

AF = FB 

Question 53

In each of the questions are question is followed by two statements I and II. The answer is

  1. if the question can be answered by one of the given statements alone and not by the other;
  2. if the question can be answered by either statement alone;
  3. if the question can be answered by both the statements together but not by any one of the two;
  4. if the question cannot be answered by using both the statement together.

Is quadrilateral ABCD a gm?

  1. Diagonal AC and BD bisect each other.
  2. Diagonal AC and BD are equal.

The correct answer is : (a)/ (b)/ (c)/ (d).

Solution 53

Correct option: (a)

If the diagonals of a quad. ABCD bisect each other, then the quad. ABCD is a parallelogram.

So, I gives the answer.

If the diagonals are equal, then the quad. ABCD is a parallelogram.

So, II gives the answer.

Question 54

In each of the questions are question is followed by two statements I and II. The answer is

  1. if the question can be answered by one of the given statements alone and not by the other;
  2. if the question can be answered by either statement alone;
  3. if the question can be answered by both the statements together but not by any one of the two;
  4. if the question cannot be answered by using both the statement together

Is quadrilateral ABCD a rhombus?

  1. Quad. ABCD is a gm.
  2. Diagonals AC and BD are perpendicular to each other.

The correct answer is: (a) / (b)/ (c)/ (d).

 

Solution 54

Correct option: (c)

If the quad. ABCD is a gm, it could be a rectangle or square or rhombus.

So, statement I is not sufficient to answer the question.

If the diagonals AC and BD are perpendicular to each other, then the gm could be a square or rhombus.

So, statement II is not sufficient to answer the question.

However, if the statements are combined, then the quad. ABCD is a rhombus.

Question 55

In each of the questions are question is followed by two statements I and II. The answer is

  1. if the question can be answered by one of the given statements alone and not by the other;
  2. if the question can be answered by either statement alone;
  3. if the question can be answered by both the statements together but not by any one of the two;
  4. if the question cannot be answered by using both the statement together

Is gm ABCD a square?

  1. Diagonals of gm ABCD are equal.
  2. Diagonals of gm ABCD intersect at right angles.

The correct answer is: (a)/ (b)/ (c)/ (d).

 

Solution 55

Correct option: (c)

If the diagonals of a gm ABCD are equal, then gm ABCD could either be a rectangle or a square.

If the diagonals of the gm ABCD intersect at right angles, then the gm ABCD could be a square or a rhombus.

However, if both the statements are combined, then gm ABCD will be a square.

Question 56

In each of the questions are question is followed by two statements I and II. The answer is

  1. if the question can be answered by one of the given statements alone and not by the other;
  2. if the question can be answered by either statement alone;
  3. if the question can be answered by both the statements together but not by any one of the two;
  4. if the question cannot be answered by using both the statement together

Is quad. ABCD a parallelogram?

  1. Its opposite sides are equal.
  2. Its opposite angles are equal.

The correct answer is: (a)/ (b)/ (c)/ (d)

 

Solution 56

Correct option: (b)

If the opposite sides of a quad. ABCD are equal, the quadrilateral is a parallelogram.

If the opposite angles are equal, then the quad. ABCD is a parallelogram.

Question 57

 Each question consists of two statements, namely, Assertion (A) and Reason (R). For selecting the correct answer, use the following code:

  1. Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
  2. Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A)
  3. Assertion (A) is true and Reason (R) is false.
  4. Assertion (A) is false and Reason (R) is true.

 

Assertion (A)

Reason (R)

 If three angles of a quadrilateral are 130°, 70° and 60°, then the fourth angle is 100°.

The sum of all the angle of a quadrilateral is 360°.

 

The correct answer is: (a)/ (b)/ (c)/ (d).

 

Solution 57

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 58

Each question consists of two statements, namely, Assertion (A) and Reason (R). for selecting the correct answer, use the following code:

  1. Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
  2. Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A)
  3. Assertion (A) is true and Reason (R) is false.
  4. Assertion (A) is false and Reason (R) is true.

 

Assertion (A)

Reason (R)

ABCD is a quadrilateral in which P, Q, R and S are the mid-points of AB, BC, CD and DA respectively.

Then, PQRS is a parallelogram.

The line segment joining the mid-points of any two sides of a triangle is parallel to the third side and equal to half of it.

 

The correct answer is: (a)/ (b)/ (c)/ (d).

 

Solution 58

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - QuadrilateralsR-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - QuadrilateralsR-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

 

The Reason (R) is true and is the correct explanation for the Assertion (A).

Question 59

Each question consists of two statements, namely, Assertion (A) and Reason (R). for selecting the correct answer, use the following code:

  1. Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
  2. Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A)
  3. Assertion (A) is true and Reason (R) is false.
  4. Assertion (A) is false and Reason (R) is true.

 

Assertion (A)

Reason (R)

In a rhombus ABCD, the diagonal AC bisects A as well as C.

The diagonals of a rhombus bisect each other at right angles.

 

The correct answer is: (a)/ (b)/ (c)/ (d).

 

Solution 59

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 60

Each question consists of two statements, namely, Assertion (A) and Reason (R). for selecting the correct answer, use the following code:

  1. Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
  2. Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A)
  3. Assertion (A) is true and Reason (R) is false.
  4. Assertion (A) is false and Reason (R) is true.

 

Assertion (A)

Reason (R)

Every parallelogram is a rectangle.

The angle bisectors of a parallelogram form a rectangle.

 

The correct answer is: (a)/ (b)/ (c)/ (d).

 

Solution 60

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 61

Each question consists of two statement, namely, Assertion (A) and Reason (R). for selecting the correct answer, use the following code:

  1. Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
  2. Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A)
  3. Assertion (A) is true and Reason (R) is false.
  4. Assertion (A) is false and Reason (R) is true.

 

Assertion (A)

Reason (R)

The diagonals of a gm bisect each other.

If the diagonals of a gm are equal and intersect at right angles, then the parallelogram is a square.

 

The correct answer is: (a)/ (b)/ (c)/ (d).

 

Solution 61

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 62

 

Column I

Column II

(a) Angle bisectors of a parallelogram form a

(p) parallelogram

(b) The quadrilateral formed by joining the mid-points of the pairs of adjacent sides of a square is a

(q) rectangle

(c) The quadrilateral formed by joining the mid-points of the pairs of adjacent side of a rectangle is a

(r) square

(d) The figure formed by joining the mid-points of the pairs of adjacent sides of a quadrilateral is

(s) rhombus

 

The correct answer is:

(a) -…….,

(b) -…….,

(c) -…….,

(d)-…….

Solution 62

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 63

 

Column I

Column II

(a) In the given figure, ABCD is a trapezium in which AB = 10 cm and CD = 7cm. If P and Q are the mid-points of AD and BC respectively, then PQ =

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

(p) equal 

(b) In the given figure, PQRS is a gm whose diagonal intersect at O. If PR = 13 cm, then OR=

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

(q) at right angle

(c) The diagonals of a square are

(r) 8.5 cm

(d) The diagonals of a rhombus bisect each other

(s) 6.5 cm

 

The correct answer is:

(a) -…….,

(b) -…….,

(c) -…….,

(d)-…….

Solution 63

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Chapter 10 - Quadrilaterals Exercise Ex. 10B

Question 1

In the adjoining figure, ABCD is a parallelogram in which R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals=720. Calculate R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals,R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilateralsand R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Solution 1

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 2

In the adjoining figure , ABCD is a parallelogram in which

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals. Calculate R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Solution 2

 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 3

In the adjoining figure, M is the midpoint of side BC of parallelogram ABCD such that BAM = DAM. Prove that AD = 2CD.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

Solution 3

ABCD is a parallelogram.

Hence, AD || BC.

DAM = AMB (alternate angles)

BAM = AMB (since BAM = DAM)

BM = AB (sides opposite to equal angles are equal)

But, AB = CD (opposite sides of a parallelogram)

BM = AB = CD ….(i)

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

Question 4

In a adjoining figure, ABCD is a parallelogram in which R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals=60o. If the parallelogram in which R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilateralsand R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilateralsmeet DC at P, prove that (i) R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - QuadrilateralsPB=90o, (ii) AD=DP and PB=PC=BC, (iii) DC=2AD.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Solution 4

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - QuadrilateralsR-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

 

 

 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

Question 5

In the adjoining figure, ABCD is a parallelogram in which

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Calculate R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Solution 5

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 6

In a ||gm ABCD , if R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals, find the value of x and the measure of each angle of the parallelogram.

Solution 6

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 7

If an angle of a parallelogram is four fifths of its adjacent angle, find the angles of the parallelogram .

Solution 7

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 8

Find the measure of each angle of parallelogram , if one of its angles is R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals less than twice the smallest angle.

Solution 8

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 9

ABCD is a parallelogram in which AB=9.5 cm and its parameter is 30 cm. Find the length of each side of the parallelogram.

Solution 9

Question 10

In each of the figures given below, ABCD is a rhombus. Find the value of x and y in each case.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Solution 10

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 11

The lengths of the diagonals of a rhombus are 24 cm and 18 cm respectively . Find the length of each side of the rhombus.

Solution 11

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 12

Each side of a rhombus is 10 cm long and one of its diagonals measures 16 cm. Find the length of the other diagonal and hence find the area of the rhombus.

Solution 12

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 13

In each of the figures given below, ABCD is a rectangle . Find the values of x and y in each case.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Solution 13

 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

 

 

Question 14

In a rhombus ABCD, the altitude from D to the side AB bisect AB. Find the angle of the rhombus

Solution 14

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

Let the altitude from D to the side AB bisect AB at point P.

Join BD.

In ΔAMD and ΔBMD,

AM = BM (M is the mid-point of AB)

AMD = BMD (Each 90°)

MD = MD (common)

ΔAMD ΔBMD (by SAS congruence criterion)

AD = BD (c.p.c.t.)

But, AD = AB (sides of a rhombus)

AD = AB = BD

ΔADB is an equilateral triangle.

A = 60° 

C = A = 60° (opposite angles are equal)

B = 180° - A = 180° - 60° = 120° 

D = B = 120° 

Hence, in rhombus ABCD, A = 60°, B = 120°, C = 60° and D = 120°.

Question 15

In the adjoining figure , ABCD is square . A line segment CX cuts AB at X and the diagonal BD at O, such that R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals. Find the value of x.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

*Back answer incorrect

Solution 15

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 16

In a rhombus ABCD show that diagonal AC bisect A as well as C and diagonal BD bisect B as well as D.

Solution 16

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

In ΔABC and ΔADC,

AB = AD (sides of a rhombus are equal)

BC = CD (sides of a rhombus are equal)

AC = AC (common)

ΔABC ΔADC (by SSS congruence criterion)

BAC = DAC and BCA = DCA (c.p.c.t.)

AC bisects A as well as C.

Similarly,

In ΔBAD and ΔBCD,

AB = BC (sides of a rhombus are equal)

AD = CD (sides of a rhombus are equal)

BD = BD (common)

ΔBAD ΔBCD (by SSS congruence criterion)

ABD = CBD and ADB = CDB (c.p.c.t.)

BD bisects B as well as D.

Question 17

In a parallelogram ABCD, points M and N have been taken on opposite sides AB and CD respectively such that AM = CN. Show that AC and MN bisect each other.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

Solution 17

In ΔAMO and ΔCNO

MAO = NCO (AB CD, alternate angles)

AM = CN (given)

AOM = CON (vertically opposite angles)

ΔAMO ΔCNO (by ASA congruence criterion)

AO = CO and MO = NO (c.p.c.t.)

AC and MN bisect each other.

Question 18

In the adjoining figure, ABCD is a parallelogram. If P and Q are points on AD and BC respectively such that R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals, prove that AQCP is a parallelogram.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Solution 18

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 19

In the adjoining figure, ABCD is a parallelogram whose diagonals intersect each other at O.A line segment EOF is drawn to meet AB at E and DC at F . Prove that OE=OF.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Solution 19

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

 

Question 20

The angle between two altitudes of a parallelogram through the vertex of an obtuse angle of the parallelogram is 60°. Find the angles of the parallelogram.

Solution 20

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

DCM = DCN + MCN

90° = DCN + 60° 

DCN = 30° 

In ΔDCN,

DNC + DCN + D = 180° 

90° + 30° + D = 180° 

D = 60° 

B = D = 60° (opposite angles of parallelogram are equal)

A = 180° - B = 180° - 60° = 120° 

C = A = 120° 

Thus, the angles of a parallelogram are 60°, 120°, 60° and 120°.

Question 21

ABCD is rectangle in which diagonal AC bisect A as well as C. Show that (i) ABCD is square, (ii) diagonal BD bisect B as well as D.

Solution 21

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

(i) ABCD is a rectangle in which diagonal AC bisects A as well as C.

BAC = DAC ….(i)

And BCA = DCA ….(ii)

Since every rectangle is a parallelogram, therefore

AB DC and AC is the transversal.

BAC = DCA (alternate angles)

DAC = DCA [From (i)]

Thus, in ΔADC,

AD = CD (opposite sides of equal angles are equal)

But, AD = BC and CD = AB (ABCD is a rectangle)

AB = BC = CD = AD

Hence, ABCD is a square.

 

(ii) In ΔBAD and ΔBCD,

AB = CD

AD = BC

BD = BD

ΔBAD ΔBCD (by SSS congruence criterion)

ABD = CBD and ADB = CDB (c.p.c.t.)

Hence, diagonal BD bisects B as well as D.

Question 22

In the adjoining figure, ABCD is a parallelogram in which AB is produced to E so that BE= AB. Prove that ED bisects BC.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Solution 22

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 23

In the adjoining figure , ABCD is a parallelogram and E is the midpoint of side BC. If DE and AB when produced meet at F, prove that AF=2AB.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Solution 23

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 24

Two parallel lines l and m are intersected by a transversal t. Show that the quadrilateral formed by the bisectors of interior angles is a rectangle.

Solution 24

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

l m and t is a transversal.

APR = PRD (alternate angles)

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

SPR = PRQ (PS and RQ are the bisectors of APR and PRD)

Thus, PR intersects PS and RQ at P and R respectively such that SPR = PRQ i.e., alternate angles are equal.

PS RQ

Similarly, we have SR PQ.

Hence, PQRS is a parallelogram.

Now, BPR + PRD = 180° (interior angles are supplementary)

2QPR + 2QRP = 180° (PQ and RQ are the bisectors of BPR and PRD)

QPR + QRP = 90° 

In ΔPQR, by angle sum property,

PQR + QPR + QRP = 180° 

PQR + 90° = 180° 

PQR = 90° 

Since PQRS is a parallelogram,

PQR = PSR

PSR = 90° 

Now, SPQ + PQR = 180° (adjacent angles in a parallelogram are supplementary)

SPQ + 90° = 180° 

SPQ = 90° 

SRQ = 90° 

Thus, all the interior angles of quadrilateral PQRS are right angles.

Hence, PQRS is a rectangle.

Question 25

K, L, M and N are point on the sides AB, BC, CD and DA respectively of a square ABCD such that AK = BL = CM = DN. Prove that KLMN is a square.

Solution 25

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

AK = BL = CM = DN (given)

BK = CL = DM = AN (i)(since ABCD is a square)

In ΔAKN and ΔBLK,

AK = BL (given)

A = B (Each 90°)

AN = BK [From (i)]

ΔAKN ΔBLK (by SAS congruence criterion)

AKN = BLK and ANK = BKL (c.p.c.t.)

But, AKN + ANK = 90° and BLK + BKL = 90° 

AKN + ANK + BLK + BKL = 90° + 90° 

2AKN + 2BKL = 180° 

AKN + BKL = 90° 

NKL = 90° 

Similarly, we have

KLM = LMN = MNK = 90° 

Hence, KLMN is a square.

Question 26

A R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilateralsis given . if lines are drawn through A, B, C parallel respectively to the sides BC, CA and AB forming R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals , as shown in the adjoining figure, show that R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Solution 26

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 27

In the adjoining figure,R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals is a triangle and through A,B, C lines are drawn , parallel respectively to BC, CA and AB, intersecting at P, Q and R. prove that the perimeter of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals is double the perimeter of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals .

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Solution 27

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

Chapter 10 - Quadrilaterals Exercise Ex. 10A

Question 1

Three angles of a quadrilateral are 75°, 90° and 75°. Find the measure of the fourth angle.

Solution 1

Let the measure of the fourth angle = x° 

For a quadrilateral, sum of four angles = 360° 

x° + 75° + 90° + 75° = 360° 

x° = 360° - 240° 

x° = 120° 

Hence, the measure of fourth angle is 120°. 

Question 2

The angle of the quadrilateral are in the ratio 2:4:5:7. Find the angles.

Solution 2

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 3

In the adjoining figure , ABCD is a trapezium in which AB || DC. If R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals=550 and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals= 700, find R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

Solution 3

Since AB || DC

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 4

In the adjoining figure , ABCD is a square andR-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilateralsis an equilateral triangle . Prove that

(i)AE=BE, (ii) R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals=150

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Solution 4

Given:R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

Question 5

In the adjoining figure , BMR-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - QuadrilateralsAC and DNR-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - QuadrilateralsAC. If BM=DN, prove that AC bisects BD.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Solution 5

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 6

In given figure, ABCD is a quadrilateral in which AB=AD and BC= DC. Prove that (i) AC bisects R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals, (ii) BE=DE,

(iii) R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Solution 6

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 7

In the given figure , ABCD is a square and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - QuadrilateralsPQR=900. If PB=QC= DR, prove that (i) QB=RC, (ii) PQ=QR, (iii) QPR=450.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Solution 7

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

 

Question 8

If O is a point with in a quadrilateral ABCD, show that OA+OB+OC+OD> AC+BD.

Solution 8

Given: O is a point within a quadrilateral ABCD

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 9

In the adjoining figure, ABCD is a quadrilateral and AC is one of the diagonals .Prove that:

(i)AB+BC+CD+DA> 2AC

(ii)AB+BC+CD>DA

(iii)AB+BC+CD+DA>AC+BD

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Solution 9

Given: ABCD is a quadrilateral and AC is one of its diagonals.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 10

Prove that the sum of all the angles of a quadrilateral is 3600.

Solution 10

Given: ABCD is a quadrilateral.

Chapter 10 - Quadrilaterals Exercise Ex. 10C

Question 1

P, Q, R and S are respectively the midpoints of the sides AB, BC, CD and DA of a quadrilateral ABCD. Show that

(i) PQ AC and PQ = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

(ii) PQ SR

(iii) PQRS is a parallelogram.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

Solution 1

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

(i) In ΔABC, P and Q are the mid-points of sides AB and BC respectively.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

(ii) In ΔADC, R and S are the mid-points of sides CD and AD respectively.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

From (i) and (ii), we have

PQ = SR and PQ SR

(iii) Thus, in quadrilateral PQRS, one pair of opposite sides are equal and parallel.

Hence, PQRS is a parallelogram. 

Question 2

A square is inscribed in an isosceles right triangle so that the square and the triangle have one angle common. Show that the vertex of the square opposite the vertex of the common angle bisect the hypotenuse.

Solution 2

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

Let ΔABC be an isosceles right triangle, right-angled at B.

AB = BC

Let PBSR be a square inscribed in ΔABC with common B.

PB = BS = SR = RP

Now, AB - PB = BC - BS

AP = CS ….(i)

In ΔAPR and ΔCSR

AP = CS  [From (i)

APR = CSR (Each 90°)

PR = SR (sides of a square)

ΔAPR ΔCSR (by SAS congruence criterion)

AR = CR (c.p.c.t.)

Thus, point R bisects the hypotenuse AC.

Question 3

In the adjoining figure , ABCD is a ||gm in which E and F are the midpoints of AB and CD respectively . If GH is a line segment that cuts AD, EF, and BC at G, P and H respectively, prove that GP = PH.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Solution 3
R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 4

M and N are points on opposites sides AD and BC of a parallelogram ABCD such that MN passes through the point of intersection O of its diagonals AC and BD. show that MN is bisected at O.

Solution 4

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

In ΔAOM and ΔCON

MAO = OCN  (Alternate angles)

AO = OC (Diagonals of a parallelogram bisect each other)

AOM = CON  (Vertically opposite angles)

ΔAOM ΔCON  (by ASA congruence criterion)

MO = NO (c.p.c.t.)

Thus, MN is bisected at point O. 

Question 5

In the adjoining figure, PQRS is a trapezium in which PQ SR and M is the midpoint of PS. A line segment MN PQ meets QR at N. Show that N is the midpoint of QR.

 

 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

Solution 5

Construction: Join diagonal QS. Let QS intersect MN at point O.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

PQ SR and MN PQ

PQ MN SR

By converse of mid-point theorem a line drawn, through the mid-point of any side of a triangle and parallel to another side bisects the third side. 

Now, in ΔSPQ

MO PQ and M is the mid-point of SP

So, this line will intersect QS at point O and O will be the mid-point of QS.

Also, MN SR

Thus, in ΔQRS, ON SR and O is the midpoint of line QS.

So, by using converse of mid-point theorem, N is the mid-point of QR.

Question 6

In a parallelogram PQRS, PQ = 12 cm and PS = 9 cm. The bisector of P meets SR in M. PM and QR both when produced meet at T. Find the length of RT.

Solution 6

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

PM is the bisector of P.

QPM = SPM ….(i)

PQRS is a parallelogram.

PQ SR and PM is the transversal.

QPM = MS (ii)(alternate angles)

From (i) and (ii),

SPM = PMS ….(iii)

MS = PS = 9 cm (sides opposite to equal angles are equal)

Now, RMT = PMS (iv)(vertically opposite angles)

Also, PS QT and PT is the transversal.

RTM = SPM

RTM = RMT

RT = RM (sides opposite to equal angles are equal)

RM = SR - MS = 12 - 9 = 3 cm

RT = 3 cm

Question 7

In the adjoining figure , ABCD is a trapezium in which AB|| DC and P, Q are the mid points of AD and BC respectively. DQ and AB when produced meet at E. Also, AC and PQ intersect at R. Prove that (i) DQ=QE, (ii) PR||AB, (iii) AR=RC.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Solution 7

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 8

In the adjoining figure, AD is a medium of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals and DE|| BA. Show that BE is also a median of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Solution 8

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 9

In the adjoining figure , AD and BE are the medians of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals and DF|| BE. Show that R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Solution 9

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 10

Prove that the line segments joining the middle points of the sides of a triangle divide it into four congruent triangles.

Solution 10

 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 11

In the adjoining figure, D,E,F are the midpoints of the sides BC, CA, and AB respectively, of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals. Show that R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Solution 11

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 12

Show that the quadrilateral formed by joining the mid points of the pairs of adjacent sides of a rectangle is a rhombus.

Solution 12

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 13

Show that the quadrilateral formed by joining the mid points of the pairs of adjacent sides of a rhombus is a rectangle.

Solution 13

 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Question 14

Show that the quadrilateral formed by joining the mid points of the pairs of adjacent sides of a square is a square.

Solution 14

 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

 

 

Question 15

Prove that the line segment joining the midpoints of opposite sides of a quadrilateral bisect each other.

Solution 15

Question 16

The diagonals of a quadrilateral ABCD are equal. Prove that the quadrilateral formed by joining the midpoints of its sides is a rhombus.

Solution 16

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

In ΔABC, P and Q are the mid-points of sides AB and BC respectively.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

In ΔBCD, Q and R are the mid-points of sides BC and CD respectively.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

In ΔADC, S and R are the mid-points of sides AD and CD respectively.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

In ΔABD, P and S are the mid-points of sides AB and AD respectively.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

PQ || RS and QR || SP [From (i), (ii), (iii) and (iv)]

Thus, PQRS is a parallelogram.

Now, AC = BD (given)

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

PQ = QR = RS = SP  [From (i), (ii), (iii) and (iv)]

Hence, PQRS is a rhombus.

Question 17

The diagonals of a quadrilateral ABCD are perpendicular to each other. Prove that the quadrilateral formed by joining the midpoints of its sides is a rectangle.

Solution 17

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

In ΔABC, P and Q are the mid-points of sides AB and BC respectively.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

In ΔADC, R and S are the mid-points of sides CD and AD respectively.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

From (i) and (ii),

PQ || RS and PQ = RS

Thus, in quadrilateral PQRS, a pair of opposite sides are equal are parallel.

So, PQRS is a parallelogram.

Let the diagonals AC and BD intersect at O.

Now, in ΔABD, P and S are the mid-points of sides AB and AD respectively.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Thus, in quadrilateral PMON, PM || NO and PN || MO.

PMON is a parallelogram.

MPN = MON (opposite angles of a parallelogram are equal)

MPN = BOA (since BOA = MON)

MPN = 90° (since AC BD, BOA = 90°)

QPS = 90° 

Thus, PQRS is a parallelogram whose one angle, i.e. QPS = 90°.

Hence, PQRS is a rectangle. 

Question 18

The midpoints of the sides AB, BC, CD and DA of a quadrilateral ABCD are joined to form a quadrilateral. If AC = BD and AC BD then prove that the quadrilateral formed is a square.

Solution 18

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals 

In ΔABC, P and Q are the mid-points of sides AB and BC respectively.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

In ΔBCD, Q and R are the mid-points of sides BC and CD respectively.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

In ΔADC, S and R are the mid-points of sides AD and CD respectively.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

In ΔABD, P and S are the mid-points of sides AB and AD respectively.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

PQ || RS and QR || SP [From (i), (ii), (iii) and (iv)]

Thus, PQRS is a parallelogram.

Now, AC = BD (given)

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

PQ = QR = RS = SP [From (i), (ii), (iii) and (iv)]

 

Let the diagonals AC and BD intersect at O.

Now,

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Quadrilaterals

Thus, in quadrilateral PMON, PM || NO and PN || MO.

PMON is a parallelogram.

MPN = MON (opposite angles of a parallelogram are equal)

MPN = BOA (since BOA = MON)

MPN = 90° (since AC BD, BOA = 90°)

QPS = 90° 

Thus, PQRS is a parallelogram such that PQ = QR = RS = SP and QPS = 90°.

Hence, PQRS is a square.