RD SHARMA Solutions for Class 9 Maths Chapter 13 - Quadrilaterals
Chapter 13 - Quadrilaterals Exercise Ex. 13.1
Question 1
Solution 1
Question 2
Solution 2
Question 3
The angles of quadrilateral are in the ratio 3: 5: 9: 13, Find all the angles of the quadrilateral.
Solution 3
Let the common ratio between the angles is x. So, the angles will be 3x, 5x, 9x and 13x respectively.
Since the sum of all interior angles of a quadrilateral is 360o.
3x + 5x + 9x + 13x = 360o
Since the sum of all interior angles of a quadrilateral is 360o.
3x + 5x + 9x + 13x = 360o30x = 360o
x = 12o
Hence, the angles are
3x = 3
12 = 36o5x = 5
12 = 60o9x = 9
12 = 108o13x = 13
12 = 156o Question 4
Solution 4
Chapter 13 - Quadrilaterals Exercise Ex. 13.2
Question 1
Two opposite angles of a parallelogram are (3x - 2)° and (50 - x)°. Find the measure of each of the parallelogram.
Solution 1
Question 2
If an angle of a parallelogram is two-third of its adjacent angle, find the angles of the parallelogram.
Solution 2
Question 3
Find the measure of all the angles of a parallelogram, if one angle is 24o less than twice the smallest angle.
Solution 3
Question 4
The perimeter of a parallelogram is 22 cm. If the longer side measures 6.5 cm what is the measure of the shorter side?
Solution 4
Question 5
In a parallelogram ABCD, ∠D = 135°, determine the measures of ∠A and ∠B.
Solution 5
Question 6
ABCD is a parallelogram in which ∠A = 70. Compute ∠B, ∠C and ∠D.
Solution 6
Question 7
In fig., ABCD is a parallelogram in which ∠DAB = 75° and ∠DBC = 60°. Compute ∠CDB and ∠ADB.
Solution 7
Question 8
Solution 8
i. F
ii. T
iii. F
iv. F
v. T
vi. F
vii. F
viii. T
Question 9
In fig., ABCD is a parallelogram in which ∠A = 60°. If the bisectors of ∠A and ∠b meet at P, prove that AD = DP, PC = BC and DC = 2AD.
Solution 9
Question 10
In fig., ABCD is a parallelogram and E is the mid-point of side BC. IF DE and AB when produced meet at F, prove that AF = 2AB.
Solution 10
Chapter 13 - Quadrilaterals Exercise Ex. 13.3
Question 1
In a parallelogram ABCD, determine sum of angles ∠C and ∠D.
Solution 1
C and D are cosecutive interior angles on the same side of the transversal CD. Therefore,
C + D = 180o
Question 2
In a parallelogram ABCD, if ∠B = 135°, determine the measures of its other angles.
Solution 2
Question 3
ABCD is a square. AC and BD intersect at O. State the measure of 
AOB.
Solution 3
Since, diagonals of a square bisect each other at right angle. Therefore, AOB = 90o
Question 4
Solution 4
Question 5
The sides AB and CD of a parallelogram ABCD are bisected at E and F. Prove that EBFD is a parallelogram.
Solution 5
Question 6
P and Q are the points of trisection of the diagonal BD of a parallelogram ABCD. Prove that CQ is parallel to AP. Prove also that AC bisects PQ.
Solution 6
Question 7
ABCD is a square E, F, G and H are points on AB, BC, CD and DA respectively, such that AE = BF = CG = DH. Prove that EFGH is square.
Solution 7
Question 8
ABCD is a rhombus, EABF is a straight line such that EA = AB = BF. Prove that ED and FC when produced meet at right angles.
Solution 8
Question 9
ABCD is a parallelogram, AD is produced to E so that DE = DC and EC produced meets AB produced in F. Prove that BF = BC.
Solution 9
Chapter 13 - Quadrilaterals Exercise Ex. 13.4
Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Question 6
Solution 6
Question 7
In fig., triangle ABC is right-angled at B. Give that AB = 9 cm, AC = 15 cm and D, E are the mid-points of the sides AB and AC respectively, calculate
(i) The length of BC
(ii) The area of 
ADE.
Solution 7
Question 8
In fig., M, N, and P are the mid-points of AB, AC and BC respectively. If MN = 3 cm, Np = 3.5 cm and MP = 2.5 cm, Calculate BC, AB and AC.
Solution 8
Question 9
In fig., AB = AC and CP ∥ BA and AP is the bisector of exterior ∠CAD of ΔABC. Prove that (i) ∠PAC = ∠BCA (ii) ABCP is a parallelogram.
Solution 9
Question 10
Solution 10
Question 11
Solution 11
Question 12
Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.
Solution 12
Let ABCD is a quadrilateral in which P, Q, R and S are mid-points of sides AB, BC, CD and DA respectively.
Join PQ, QR, RS, SP and BD.
In
ABD, S and P are mid points of AD and AB respectively.
So, By using mid-point theorem, we can say that
SP || BD and SP =
BD ... (1)
Similarly inBCD
QR || BD and QR = BD ... (2)
From equations (1) and (2), we have
SP || QR and SP = QR
As in quadrilateral SPQR one pair of opposite sides are equal and parallel to
each other.
So, SPQR is a parallelogram.
Join PQ, QR, RS, SP and BD.
In
ABD, S and P are mid points of AD and AB respectively. So, By using mid-point theorem, we can say that
SP || BD and SP =
BD ... (1) Similarly in
BCDQR || BD and QR =
BD ... (2) From equations (1) and (2), we have
SP || QR and SP = QR
As in quadrilateral SPQR one pair of opposite sides are equal and parallel to
each other.
So, SPQR is a parallelogram.
Since, diagonals of a parallelogram bisect each other.
Hence, PR and QS bisect each other.
Hence, PR and QS bisect each other.
Question 13
Fill in the blanks to make the following statements correct:
(i) The triangle formed by joining the mid-points of the sides of an isosceles traingle is ______.
(ii) The triangle formed by joining the mid-points of the sides of a right triangle is ______ .
(iii) The figure formed by joining the mid-points of consecutive sides of a quadrilateral is ______ .
Solution 13
(i) isosceles
(ii) right triangle
(iii) parallelogram
Question 14
Solution 14
Question 15
In fig., BE ⊥ AC. AD is any line from A to BC interesting BE in H. P, Q and R are respectively the mid-points of AH, AB and BC. Prove that ∠PQR = 90°.
Solution 15
Question 16
Solution 16
Question 17
In fig., ABCD is a parallelogram in which P is the mid-point of DC and Q is a point on AC such that CQ = (1/4)AC. If PQ produced meets BC at R, prove that R is a mid-point of BC.
Solution 17
Question 18
In fig., ABCD and PQRC are rectangle and Q is the mid-point of AC. Prove that
i. DP = PC ii. PR = (1/2) AC
Solution 18
Question 19
ABCD is a parallelogram, E and F are the mid-points of AB and CD respectively. GH is any line intersecting AD, EF and BC and G, P and H respectively. Prove that GP = PH.
Solution 19
Question 20
BM and CN are perpendiculars to a line passing through the vertex A of a triangle ABC. If L is the mid-point of BC, prove that LM = LN.
Solution 20
Chapter 13 - Quadrilaterals Exercise 13.70
Question 1
The opposite sides of a quadrilateral have
(a) no common point
(b) one common point
(c) two common points
(d) infinitely many common points
Solution 1
ABCD is a Quadrilateral.
The opposite sides AB and DC, AD and BC have no common point.
Hence, correct option is (a).
Question 2
The consecutive sides of a quadrilateral have
(a) no common point
(b) one common point
(c) two common points
(d) infinitely many common points
Solution 2
Consecutive sides of a Quadrilateral ABCD are
AB and BC,
BC and CD,
CD and AD,
AD and AB,
which have only one point in common
i.e the joint point of their ends.
Hence, correct option is (b).
Chapter 13 - Quadrilaterals Exercise 13.71
Question 3
Solution 3
Question 4
Which of the following quadrilateral is not a rhombus?
(a) All four sides are equal
(b) Diagonals bisect each other
(c) Diagonals bisect opposite angles
(d) One angle between the diagonals is 60°
Solution 4
For a rhombus, the angle between the diagonals is 90° and not 60°.
Hence, correct option is (d).
Question 5
Diagonals necessarily bisect opposite angles in a
(a) rectangle
(b) parallelogram
(c) isosceles trapezium
(d) square
Solution 5
Diagonals necessarily bisect opposite angles in a square.
Hence, correct option is (d).
Question 6
The two diagonals are equal in a
(a) parallelogram
(b) rhombus
(c) rectangle
(d) trapezium
Solution 6
The two diagonals are equal in a rectangle (property).
Hence, correct option is (c).
Question 7
We get a rhombus by joining the mid-points of the sides of a
(a) parallelogram
(b) rhombus
(c) rectangle
(d) triangle
Solution 7
Question 8
The bisectors of any two adjacent angles of a parallelogram intersect at
(a) 30°
(b) 45°
(c) 60°
(d) 90°
Solution 8
Question 9
The bisectors of the angle of a parallelogram enclose a
(a) parallelogram
(b) rhombus
(c) rectangle
(d) square
Solution 9
AR, BR, CP, DP are the bisectors of angles of parallelogram.
Because two bisectors of adjacent angles make 90° between them So PQRS is a Rectangle
Because DP and BR are acute angle bisectors so the distance between them PQ < PS (The distance between other two bisectors)
So PQ ≠ PS (So PQRS is not a square, but only a rectangle)
Hence, correct option is (c).
Question 10
The figure formed by joining the mid-points of the adjacent sides of a quadrilateral is a
(a) parallelogram
(b) rectangle
(c) square
(d) rhombus
Solution 10
Question 11
The figure formed by joining the mid-points of the adjacent sides of a rectangle is a
(a) square
(b) rhombus
(c) trapezium
(d) none of these
Solution 11
Question 12
The figure formed by joining the mid-points of the adjacent sides of a rhombus is a
(a) square
(b) rectangle
(c) trapezium
(d) none of these
Solution 12
Question 13
The figure formed by joining the mid-points of the adjacent sides of a square is a
(a) rhombus
(b) square
(c) rectangle
(d) parallelogram
Solution 13
Question 14
The figure formed by joining the mid-points of the adjacent sides of a parallelogram is a
(a) rectangle
(b) parallelogram
(c) rhombus
(d) square
Solution 14
Question 15
If one angle of a parallelogram is 24° less than twice the smallest angle, then the measure of the largest angle of the parallelogram is
(a) 176°
(b) 68°
(c) 112°
(d) 102°
Solution 15
Question 16
Solution 16
Question 17
ABCD is a parallelogram and E and F are the centroids of triangles ABD and BCD respectively, then EF =
(a) AE
(b) BE
(c) CE
(d) DE
Solution 17
Chapter 13 - Quadrilaterals Exercise 13.72
Question 18
Solution 18
Question 19
If an angle of a parallelogram is is two-third of its adjacent angle, the smallest angle of the parallelogram is
(a) 108°
(b) 54°
(c) 72°
(d) 81°
Solution 19
Question 20
If the degree measures of the angles of quadrilateral are 4x, 7x, 9x and 10x, what is the sum of the measures of the smallest angle and largest angle?
(a) 140°
(b) 150°
(c) 168°
(d) 180°
Solution 20
Sum of all angles of a Quadrilateral = 360°
4x + 7x + 9x + 10x = 360°
30x = 360°
x = 12°
So, sum of smallest and largest angle,
i.e. 4x + 10x = 14x = 14 × 12 = 168°
Hence, correct option is (c).
Question 21
If the diagonals of a rhombus are 18 cm and 24 cm respectively, then its side is equal to
(a) 16 cm
(b) 15 cm
(c) 20 cm
(d) 17 cm
Solution 21
Question 22
Solution 22
Question 23
Solution 23
Question 24
Solution 24
Question 25
Solution 25
Question 26
Solution 26
Question 30
Solution 30
Question 27
Solution 27
Question 28
Solution 28
Question 29
In ΔABC, E is the mid-point of median AD such that BE produced meets AC at F. If AC = 10.5 cm, then AF =
(a) 3 cm
(b) 3.5 cm
(c) 2.5 cm
(d) 5 cm
Solution 29
Chapter 13 - Quadrilaterals Exercise 13.73
Question 31
Solution 31
Question 32
Solution 32
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