Chapter 13 : Quadrilaterals - Rd Sharma Solutions for Class 9 Maths CBSE

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Chapter 13 - Quadrilaterals Excercise 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

30x = 360o
    x = 12o
Hence, the angles are
3x = 3  12 = 36o
5x = 5  12 = 60o
9x = 9  12 = 108o
13x = 13  12 = 156o

Question 4

Solution 4


Chapter 13 - Quadrilaterals Excercise 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 Excercise 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 Excercise 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 in BCD
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.
Since, diagonals of a parallelogram 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 Excercise 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 Excercise 13.71

Question 1

begin mathsize 12px style PQRS space is space straight a space quadrilateral. space PR space and space QS space intersect space each space other space at space straight O. space In space which space of space the space following space cases comma
PQRS space is space parallelogram ?
left parenthesis straight a right parenthesis space angle straight P space equals space 100 degree comma space angle straight Q space equals space 80 degree comma space angle straight R space equals space 100 degree
left parenthesis straight b right parenthesis space angle straight P space equals space 85 degree comma space angle straight Q space equals space 85 degree comma space angle straight R space equals space 95 degree
left parenthesis straight c right parenthesis space PQ space equals space 7 space cm comma space QR space equals space 7 space cm comma space RS space equals space 8 space cm comma space SP space equals space 8 space cm
left parenthesis straight d right parenthesis space OP space equals space 6.5 space cm comma space OQ space equals space 6.5 space cm comma space OR space equals space 5.2 space cm comma space OS space equals space 5.2 space cm end style

 

Solution 1

begin mathsize 12px style In space straight a space parallelogram comma space opposite space corner space angles space are space equal space and space sum space of space adjacent space angles space equals space 180 degree
Hence comma space in space quadrilateral space PQRS comma
rightwards double arrow angle straight P space equals space angle straight R space and space angle straight Q space equals space angle straight S
Also comma space angle straight P space plus space angle straight Q space equals angle straight Q space plus space angle straight R space equals space 180 degree
Hence comma space if space angle straight P equals 100 degree space and space angle straight Q equals 80 degree comma space then space
angle straight P plus angle straight Q equals 100 degree plus 80 degree equals 180 degree
Also comma space if space angle straight Q equals 80 degree space and space angle straight R equals 100 degree comma space then
angle straight Q plus angle straight R equals 80 degree plus 100 degree equals 180 degree
Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 2

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 2

For a rhombus, the angle between the diagonals is 90° and not 60°.

Hence, correct option is (d).

Question 3

Diagonals necessarily bisect opposite angles in a

(a) rectangle

(b) parallelogram

(c) isosceles trapezium

(d) square

Solution 3

Diagonals necessarily bisect opposite angles in a square.

Hence, correct option is (d).

Question 4

The two diagonals are equal in a 

(a) parallelogram

(b) rhombus

(c) rectangle

(d) trapezium

Solution 4

The two diagonals are equal in a rectangle (property).

Hence, correct option is (c).

Question 5

We get a rhombus by joining the mid-points of the sides of a 

(a) parallelogram

(b) rhombus

(c) rectangle

(d) triangle

Solution 5

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Error converting from MathML to accessible text.

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begin mathsize 12px style By space joining space the space mid minus points space of space sides space of space straight a space triangle comma space no space quadrilateral space is space formed.

hence comma space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 6

The bisectors of any two adjacent angles of a parallelogram intersect at

(a) 30°

(b) 45°

(c) 60°

(d) 90°

Solution 6

begin mathsize 12px style In space straight a space parallelogram comma space sum space of space adjacent space angles space equals space 180 degree
rightwards double arrow angle straight A space plus space angle straight B space equals space 180 degree
rightwards double arrow fraction numerator angle straight A over denominator 2 end fraction plus fraction numerator angle straight B over denominator 2 end fraction equals 90 degree space space space space.... open parentheses 1 close parentheses
rightwards double arrow angle OAB space equals space fraction numerator angle straight A over denominator 2 end fraction space and space angle OBA space equals space fraction numerator angle straight B over denominator 2 end fraction
Thus comma space angle OAB space plus space angle OBA space equals space 90 degree space space space space space space space space space space space space space space left square bracket From space eq space open parentheses 1 close parentheses right square bracket
rightwards double arrow angle AOB space equals space 180 degree space minus space open parentheses angle OAB space plus space angle OBA close parentheses equals 180 degree space minus space 90 degree
rightwards double arrow angle AOB space equals space 90 degree
Hence comma space correct space option space is space left parenthesis straight d right parenthesis. end style

Question 7

The bisectors of the angle of a parallelogram enclose a

(a) parallelogram

(b) rhombus

(c) rectangle

(d) square

Solution 7

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 8

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 8

begin mathsize 12px style straight P comma space straight Q comma space straight R space & space straight S space are space the space mid minus points space of space AB comma space BC comma space CD space & space AD space respectively.
Consider space triangle ADB comma
If space in space straight a space triangle comma space the space mid minus points space of space two space sides space are space joint space by space straight a space line space then space the
line space is space parallel space to space the space third space side.
rightwards double arrow PS space parallel to space DB space in space triangle ADB space space space
Similiarly space in space triangle CDB comma
RQ space parallel to space DB
Hence space PS space parallel to space RQ space space space space space space space space... open parentheses 1 close parentheses
Similarly space in space triangle ABC space and space triangle ADC
SR space parallel to space AC comma space space space space space PQ space parallel to space AC
rightwards double arrow SR space parallel to space PQ space space space... open parentheses 2 close parentheses
From space eq. space open parentheses 1 close parentheses space and space open parentheses 2 close parentheses comma space space PQRS space is space straight a space parallelogram.
Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 9

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 9

begin mathsize 12px style PQ space parallel to space AC space space space left parenthesis since space in space triangle ABC space mid minus points space of space AB space & space BC space are space meeting space by space PQ right parenthesis
Similiarly comma space SR space parallel to space AC
rightwards double arrow PQ space parallel to space SR
Now space in space triangle ABD space and space triangle CBD comma
PS space parallel to space BD space and space QR space parallel to space BD
rightwards double arrow PS space parallel to space QR
Hence comma space PQRS space is space straight a space parallelogram.
But space PR space perpendicular space QS
rightwards double arrow Diagonals space cut space at space 90 degree
rightwards double arrow PQRS thin space is space straight a space Rhomus
Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 10

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 10

begin mathsize 12px style In space triangle ABD space and space triangle CBD
PS space parallel to space BD space and space QR space parallel to space BD
open curly brackets straight A space line space joining space mid minus points space of space two space sides space of space triangle space is space parallel space to space third space side close curly brackets
rightwards double arrow PS space parallel to space QR
Similiarly space PQ space parallel to space SR
Because space SR space parallel to space AC space and space QR space parallel to space BD comma
And space angle space between space the space diagonals space of space straight a space Rhombus space AC space and space BD space equals space 90 degree comma
Angle space between space SR space and space QR space equals space 90 degree
rightwards double arrow PQRS space is space straight a space Rectangle.
Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 11

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 11

begin mathsize 12px style PS space parallel to space QR comma space space space space space PQ space parallel to space SR space space space.... open parentheses 1 close parentheses space
open curly brackets Because space lines space joining space the space midpoint space of space any space two space sides space of space straight a space triangle are space parallel space to space the space third space side close curly brackets
AC space perpendicular space BD space & space PR space perpendicular space QS space space space space space space left parenthesis From space space Figure right parenthesis
SR space parallel to space AC space and space QR space parallel to BD
AC space perpendicular space BD
rightwards double arrow SR space perpendicular space QR
Hence space angle SRQ space equals space 90 degree space space space.... open parentheses 2 close parentheses
Also space triangle APS space approximately equal to space triangle DSR
rightwards double arrow PS space equals SR space space... open parentheses 3 close parentheses
From space equations space open parentheses 1 close parentheses comma space open parentheses 2 close parentheses comma space open parentheses 3 close parentheses
PQRS space is space straight a space square.
Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 12

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 12

begin mathsize 12px style PQ space parallel to space SR space parallel to space AC
QR space parallel to space PS space parallel to space BD
open curly brackets Because space line space joining space the space mid minus points space of space two space sides space of space triangle space is space parallel to space to space third space side close curly brackets
Now space because space AC space is space not space prependicular space to space BD space in space parallelogram comma
rightwards double arrow SR space is space not space perpendicular space to space QR
Also space triangle ASP space ≇ space triangle DRS
rightwards double arrow PS space not equal to space SR
rightwards double arrow PQRS space is space just space straight a space parallelogram.
Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 13

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 13

begin mathsize 12px style Let space the space smallest space angle space equals space angle ADC space equals space straight x degree
Other space angle space equals space angle BCD
rightwards double arrow angle BCD space equals space 2 straight x degree space minus space 24 degree
Also comma space angle ADC space plus space angle BCD space equals space 180 degree space space open parentheses Sum space of space adjacent space angles space in space parallel to to the power of gram space equals space 180 degree close parentheses
rightwards double arrow straight x degree space plus space 2 straight x degree space minus space 24 degree space equals space 180 degree
rightwards double arrow 3 straight x degree space space equals space 204 degree
rightwards double arrow straight x space equals space 68 degree
rightwards double arrow Largest space angle space equals space angle BCD space equals space 2 cross times 68 degree minus 24 degree space equals space 112 degree
Hence comma space correct space option space is space left parenthesis straight c right parenthesis.
end style

Question 14

begin mathsize 12px style In space straight a space parallelogram space ABCD comma space if space angle DAB space equals space 75 degree space and space angle DBC space equals space 60 degree comma space then space angle BDC space equals
left parenthesis straight a right parenthesis space 75 degree
left parenthesis straight b right parenthesis space 60 degree
left parenthesis straight c right parenthesis space 45 degree
left parenthesis straight d right parenthesis space 55 degree end style

Solution 14


begin mathsize 12px style In space parallelogram space ABCD comma
angle straight A space plus space angle straight D space equals space 180 degree
rightwards double arrow angle straight D space equals space 180 degree space minus space 75 degree space equals 105 degree
angle ADB space equals space angle DBC space left parenthesis Alternate space angles right parenthesis
rightwards double arrow angle ADB space equals space 60 degree
angle BDC space equals space angle ADC space minus space angle ADB space equals space 105 degree space minus space 60 degree equals 45 degree
Hence comma space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 15

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 15

begin mathsize 12px style Centroid space is space the space point space where space all space medians space of space straight a space triangle space meet.
In space triangle ABD comma space straight E space is space the space centroid comma
and space in space triangle BCD comma space straight F space is space the space centroid.
By space the space property space of space centroid comma space centroid space divides space straight a space median space in space 2 space colon space 1 space
So space from space figure comma
AE over EO equals 2 over 1 rightwards double arrow EO equals AE over 2 space space... open parentheses 1 close parentheses
Also space CF over FO equals 2 over 1 rightwards double arrow space FO equals CF over 2 space... open parentheses 2 close parentheses
Because space AC space is space straight a space diagonal space of space straight a space paralleogram comma space straight O space is space its space midpoint.
rightwards double arrow OA space equals space OC
rightwards double arrow AE space equals space CF
Adding space equations space open parentheses 1 close parentheses space & space open parentheses 2 close parentheses comma
EO space plus space FO space equals space fraction numerator AE space plus space CF over denominator 2 end fraction equals fraction numerator up diagonal strike 2 AE over denominator up diagonal strike 2 end fraction
rightwards double arrow EF space equals fraction numerator up diagonal strike 2 AE over denominator up diagonal strike 2 end fraction
rightwards double arrow EF space equals space AE
Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

Chapter 13 - Quadrilaterals Excercise 13.72

Question 1

begin mathsize 12px style ABCD space is space straight a space parallelogram comma space straight M space is space the space mid minus point space of space BD space and space BM space bisects space angle straight B. space Then comma space angle AMB space equals
left parenthesis straight a right parenthesis space 45 degree
left parenthesis straight b right parenthesis space 60 degree
left parenthesis straight c right parenthesis space 90 degree
left parenthesis straight d right parenthesis space 75 degree end style

Solution 1

begin mathsize 12px style angle ABM space equals space angle CBM space space... open parentheses 1 close parentheses space left parenthesis BM space bisects space angle straight B right parenthesis
angle ABM space equals space angle MDC space space... open parentheses 2 close parentheses space space space left parenthesis Alternate space angles right parenthesis
angle CBM space equals space angle ADM space space.... open parentheses 3 close parentheses space space space left parenthesis Alternate space angles right parenthesis
From space equations space open parentheses 1 close parentheses comma space open parentheses 2 close parentheses space & space open parentheses 3 close parentheses
angle MDC space equals space angle ADM space space space.... open parentheses 4 close parentheses
Now comma space consider space triangle ABD space & space triangle CBD
angle CBD space equals space angle ABD space space space space space space space open curly brackets From space eq space open parentheses 1 close parentheses close curly brackets
DB space equals space DB space left parenthesis Common right parenthesis
angle ADB space equals space angle CDB space space space space space space space space open curly brackets From space eq space open parentheses 4 close parentheses close curly brackets
Hence comma space by space ASA space property comma
triangle ABD space approximately equal to space triangle CBD
rightwards double arrow AB space equals space CB comma space AD space equals space CD
Hence comma space it space becomes space straight a space Rhombus.
So comma space now space diagonals space of space straight a space Rhombus space bisect space each space other space at space 90 degree.
rightwards double arrow angle AMB space equals space 90 degree
Hence comma space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 2

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 2

begin mathsize 12px style Let space ABCD space be space straight a space parallelogram space and space angle straight A space equals space 2 over 3 angle straight B
Also comma space angle straight A space plus space angle straight B space equals 180 degree space left parenthesis adjacent space angles space in space straight a space Parallelogram space are space supplementry right parenthesis
rightwards double arrow 2 over 3 angle straight B space plus space angle straight B space equals space 180 degree
rightwards double arrow angle straight B space equals space 108 degree space and space space angle straight A space equals space 72 degree
rightwards double arrow Smallest space angle space is space 72 degree.
Hence comma space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 3

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 3

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 4

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 4

begin mathsize 12px style Let space BD space equals space 24 space cm space and space AC space equals space 18 space cm space left parenthesis Given right parenthesis
Now comma space AO space equals AC over 2 equals 18 over 2 equals 9 space cm space and space BO equals BD over 2 equals 24 over 2 equals 12 space cm
Now comma space AB space equals square root of open parentheses AO close parentheses squared plus open parentheses BO close parentheses squared end root space space space space space left parenthesis Diagonals space make space 90 degree space between space them right parenthesis
space space space space space space space space space space space space space space space space space equals square root of 9 squared plus 12 squared end root
space space space space space space space space space space space space space space space space space equals square root of 81 space plus space 144 end root
space space space space space space space space space space space space space space space space space equals square root of 225
rightwards double arrow AB space equals space 15 space cm
Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 5

begin mathsize 12px style ABCD space is space straight a space parallelogram space in space which space diagonal space AC space bisects space angle BAD. space If space angle BAC space equals space 35 degree comma space then space angle ABC space equals
left parenthesis straight a right parenthesis space 70 degree
left parenthesis straight b right parenthesis space 110 degree
left parenthesis straight c right parenthesis space 90 degree
left parenthesis straight d right parenthesis space 120 degree end style

Solution 5

begin mathsize 12px style AC space bisects space angle DAB.
rightwards double arrow angle DAC space equals space angle BAC space equals 35 degree
rightwards double arrow angle BAD space equals space 2 space cross times space 35 degree space equals space 70 degree
angle straight A space plus space angle straight B space equals space 180 degree space space left parenthesis Sum space of space any space two space adjacent space angles space in space Parallelogram equals 180 degree right parenthesis
rightwards double arrow angle straight B space equals angle ABC space equals space 180 degree space minus space angle BAD equals 180 degree space minus space 70 degree equals 110 degree
Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 6

begin mathsize 12px style In space straight a space rhombus space ABCD comma space if space angle ACB space equals space 40 degree comma space then space angle ADB space equals
left parenthesis straight a right parenthesis space 70 degree
left parenthesis straight b right parenthesis space 45 degree
left parenthesis straight c right parenthesis space 50 degree
left parenthesis straight d right parenthesis space 60 degree end style

Solution 6

begin mathsize 12px style Consider space triangle AOD space & space triangle COB
angle AOD space equals space angle COB space equals space 90 degree
AD space equals space BC space left parenthesis Sides space of space Rhombus right parenthesis
AO space equals space CO space left parenthesis Diagonals space bisects space each space other right parenthesis
So space by space RHS space property comma space triangle AOD space approximately equal to space triangle COB
rightwards double arrow angle OAD equals angle OCB space equals space 40 degree
In space triangle AOD comma
angle ADB space equals space angle ADO space equals space 180 degree space minus space 90 degree space minus space 40 degree equals 50 degree
Hence comma space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 7

begin mathsize 12px style In space triangle ABC comma space angle straight A space equals space 30 degree comma space angle straight B space equals space 40 degree space and space angle straight C space equals space 110 degree. space The space angles space of space the space triangle space formed space by
joining space the space mid minus points space of space the space sides space of space this space triangle space are
left parenthesis straight a right parenthesis space 70 degree comma space 70 degree comma space 40 degree
left parenthesis straight b right parenthesis space 60 degree comma space 40 degree comma space 80 degree
left parenthesis straight c right parenthesis space 30 degree comma space 40 degree comma space 110 degree
left parenthesis straight d right parenthesis space 60 degree comma space 70 degree comma space 50 degree end style

Solution 7

begin mathsize 12px style If space in space any space triangle comma space all space the space mid minus points space left parenthesis of space each space sides right parenthesis space are space joined space to space form space straight a space triangle comma
then space that space triangle space is space similian space to space straight a space parent space triangle.
straight i. straight e. space triangle QPR space tilde space triangle ABC
So space angles space of space triangle PQR space will space be space same space as space angles space of space triangle ABC.
Thus comma space angles space are space 30 degree comma space 40 degree comma space 110 degree.
Hence comma space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 8

begin mathsize 12px style The space diagonals space of space straight a space parallelogram space ABCD space intersect space at space straight O. space If space angle BOC space equals space 90 degree space and
angle BDC space equals space 50 degree comma space then space angle OAB space equals
left parenthesis straight a right parenthesis space 40 degree
left parenthesis straight b right parenthesis space 50 degree
left parenthesis straight c right parenthesis space 10 degree
left parenthesis straight d right parenthesis space 90 degree end style

Solution 8

begin mathsize 12px style In space straight a space parallelogram space ABCD comma
angle OAB space equals space angle OCD
In space triangle OCD
angle OCD space plus space angle COD space plus space angle ODC space equals space 180 degree
angle COD space equals space 90 degree
angle ODC space equals 50 degree space left parenthesis given right parenthesis
angle OCD space equals space 180 degree space minus space 90 degree space minus space 50 degree space equals space 40 degree
rightwards double arrow angle OAB space equals space angle OCD space equals space 40 degree
Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 9

begin mathsize 12px style ABCD space is space straight a space trapezium space in space which space AB space parallel to space DC. space straight M space and space straight N space are space the space mid minus points space of space AD space and space BC space respectively. space
If space AB space equals space 12 space cm comma space MN space equals space 14 space cm comma space then space CD space equals
left parenthesis straight a right parenthesis space 10 space cm
left parenthesis straight b right parenthesis space 12 space cm
left parenthesis straight c right parenthesis space 14 space cm
left parenthesis straight d right parenthesis space 16 space cm end style

Solution 9

begin mathsize 12px style Let space straight a space line space BP space is space drawn space parallel to space to space AD space to space meet space DC space at space straight P.
ABPD space is space straight a space parallelogram.
AB space parallel to space PD comma space space AD space parallel to space BP
So space AB space equals space DP
Let space BP space cuts space MN space at space straight Q.
MQ space is space also space parallel to space to space AB space parallel to space PD
So space AB space equals space MQ space equals space PD space equals space 12 space cm space space.... open parentheses 1 close parentheses
QN space equals space MN space minus space MQ space equals space 14 space minus space 12 space equals space 2 space cm
Consider space triangle BPC.
straight Q space and space straight N space are space the space mid minus points space of space BP space & space BC comma space and space the space line space joining space them space QN space parallel to space PC.
Then space by space property comma space QN over PC equals 1 half
rightwards double arrow PC space equals space 2 QN space equals space 2 space cross times space 2 space equals space 4 space cm
Now comma space DC equals DP plus PC
DP space equals space 12 space cm space left square bracket From space left parenthesis 1 right parenthesis right square bracket
rightwards double arrow DC equals 12 plus 4 equals 16 space cm
Hence comma space correct space option space is space left parenthesis straight d right parenthesis. end style

Question 10

begin mathsize 12px style ABCD space is space straight a space parallelogram comma space and space straight E space is space the space mid minus point space of space BC.
DC space and space AB space when space produced space meet space at space straight F. space Then comma space AF space equals
left parenthesis straight a right parenthesis space 3 over 2 AB
left parenthesis straight b right parenthesis space 2 space AB
left parenthesis straight c right parenthesis space 3 space AB
left parenthesis straight d right parenthesis space 5 over 4 AB
end style

Solution 10

begin mathsize 12px style BC space parallel to space AD
rightwards double arrow BE space parallel to space AD
Now comma space consider space triangle FAD
BE space parallel to space AD
straight A lso space BE over AD equals 1 half
In space triangle FBE space and space triangle FAD comma space
angle FAD space equals space angle FBE space space space space space open curly brackets Corresponding space angles close curly brackets
angle ADF space equals space angle BEF space space space space space open curly brackets Corresponding space angles close curly brackets
angle straight F space equals space angle straight F space space space space space space space left curly bracket common right parenthesis
Hence comma space triangle FBE space tilde space triangle FAD
rightwards double arrow BF over AF equals BE over AD equals 1 half
rightwards double arrow 1 minus BF over AF equals 1 minus 1 half
rightwards double arrow fraction numerator AF minus BF over denominator AF end fraction equals 1 half
rightwards double arrow AB over AF equals 1 half
rightwards double arrow AF space equals space 2 AB
Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 11

begin mathsize 12px style Diagonals space of space straight a space quadrilateral space ABCD space bisect space each space other. space If space angle straight A space equals space 45 degree comma space then space angle straight B space equals
left parenthesis straight a right parenthesis space 115 degree
left parenthesis straight b right parenthesis space 120 degree
left parenthesis straight c right parenthesis space 125 degree
left parenthesis straight d right parenthesis space 135 degree end style

Solution 11

begin mathsize 12px style Consider space triangle AOD space & space triangle COB comma
AO space equals space CO space open curly brackets Diagonals space Bisects space each space other close curly brackets
OD space equals space OB space open curly brackets Diagonals space Bisects space each space other close curly brackets space space
angle AOD space equals space angle COB space left parenthesis opposite space angles right parenthesis
So comma space by space SAS space property comma space triangle AOD space approximately equal to space triangle COB
rightwards double arrow angle ADO space equals space angle CBO space space... open parentheses 1 close parentheses
angle ABD space equals space 180 degree space minus space angle straight A space minus space angle ADO space space space space space left parenthesis in space triangle ADB right parenthesis
space space space space space space space space space space space space space equals 180 space minus space 45 degree space minus space angle ADO
angle ABD space equals space 135 degree space minus space angle ADO space space space... open parentheses 2 close parentheses
angle straight B space equals space angle ABD space plus space angle CBO
Putting space values space From space eq space open parentheses 1 close parentheses space and space open parentheses 2 close parentheses
angle straight B space equals space 135 degree space minus space up diagonal strike angle ADO end strike space plus space up diagonal strike angle ADO end strike
angle straight B space equals space 135 degree
Hence comma space correct space option space is space left parenthesis straight d right parenthesis. end style

Question 12

begin mathsize 12px style straight P space is space the space mid minus point space of space side space BC space of space straight a space parallelogram space ABCD space such space that
angle BAP space equals space angle DAP. space If space AD space equals space 10 space cm comma space then space CD space equals
left parenthesis straight a right parenthesis space 5 space cm
left parenthesis straight b right parenthesis space 6 space cm
left parenthesis straight c right parenthesis space 8 space cm
left parenthesis straight d right parenthesis space 10 space cm end style

Solution 12

begin mathsize 12px style Let space straight a space line space parallel space to space AB space is space drawn space from space straight P space to space meet space AD space at space straight Q.
PQ space parallel to space AB space parallel to space DC
straight Q space is space also space mid minus point space of space AD.
Now comma space consider space parallelogram space ABPQ.
angle PAQ space equals space angle APB space left parenthesis Alternate space angles right parenthesis
Also space angle PAQ space equals space angle BAP space left parenthesis Given right parenthesis
rightwards double arrow angle APB space equals space angle BAP
So space triangle ABP space is space isosceles space triangle.
rightwards double arrow BP space equals space AB
straight i. straight e. space AB space equals space 10 over 2 equals space 5 space cm
Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 13

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 13

begin mathsize 12px style straight A space line space DG space is space drawn space parallel space to space FE space to space meet space AC.
FE space parallel to space DG space and space FE space parallel to space GH
Now comma space consider space triangle ADG.
straight E space is space the space mid minus point space of space AD space and space EF space is space line space From space straight E space parallel to space to space Base space DG.
So space by space Property comma space it space will space meet space AG space at space its space midpoint space
straight i. straight e. space straight F space is space midpoint space of space AG.
rightwards double arrow AF space equals space FG space space..... open parentheses 1 close parentheses
Now comma space consider space triangle FBC space & space triangle GDC
FE space parallel to space GH space and space FE space parallel to GD
straight D space is space mid minus point space of space BC.
rightwards double arrow DC over BC equals 1 half space space space space space space space space.... left parenthesis 2 right parenthesis
Because space triangle FBC space tilde space triangle GDC comma
rightwards double arrow GC over FC equals 1 half rightwards double arrow FC space equals space 2 GC
OR space space FG space equals space GC space space space space..... open parentheses 3 close parentheses
From space equations space open parentheses 1 close parentheses space and space open parentheses 3 close parentheses comma
AF space equals space FG space equals space GC
rightwards double arrow AF space equals space AC over 3 equals fraction numerator 10.5 over denominator 3 end fraction equals 3.5 space cm
Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Chapter 13 - Quadrilaterals Excercise 13.73

Question 1

begin mathsize 12px style In space straight a space quadrilateral space ABCD comma space angle straight A space plus space angle straight C space is space 2 space times space angle straight B space plus space angle straight D. space space If space angle straight A space equals space 140 degree space and space angle straight D equals 60 degree comma space then space angle straight B space equals
left parenthesis straight a right parenthesis space 60 degree
left parenthesis straight b right parenthesis space 80 degree
left parenthesis straight c right parenthesis space 120 degree
left parenthesis straight d right parenthesis space None space of space these end style

Solution 1

begin mathsize 12px style In space straight a space quadrilateral space ABCD comma
angle straight A space plus space angle straight B space plus space angle straight C space plus space angle straight D space equals space 360 degree space space space space space space... open parentheses 1 close parentheses
Now comma space angle straight A space plus space angle straight C space equals space 2 left parenthesis angle straight B space plus space angle straight D right parenthesis space space space space space space space left parenthesis given right parenthesis space.... open parentheses 2 close parentheses
Also comma space angle straight A space equals space 140 degree space space space angle straight D space equals space 60 degree
Putting space value space of space left parenthesis angle straight A space plus space angle straight C right parenthesis space from space eq. space open parentheses 2 close parentheses space in space eq. space open parentheses 1 close parentheses
2 open parentheses angle straight B space plus space angle straight D close parentheses space plus space angle straight B space plus space angle straight D space equals space 360 degree
3 open parentheses angle straight B space plus space angle straight D close parentheses space equals space 360 degree
rightwards double arrow angle straight B space plus space angle straight D space equals space 120 degree
rightwards double arrow angle straight B space plus space 60 degree space equals space 1200 degree
rightwards double arrow angle straight B space equals space 60 degree
Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 2

begin mathsize 12px style The space diagonals space AC space and space BD space of space straight a space rectangle space ABCD space intersect space each space other space at space straight P. space If space angle ABD space equals space 50 degree comma space then space angle DPC space equals
left parenthesis straight a right parenthesis space 70 degree
left parenthesis straight b right parenthesis space 90 degree
left parenthesis straight c right parenthesis space 80 degree
left parenthesis straight d right parenthesis space 100 degree
end style

Solution 2

begin mathsize 12px style In space triangle ABD comma
angle BDA space plus space angle ABD space plus space angle DAB space equals space 180 degree
angle ABD space equals space 50 degree space and space angle DAB space equals space 90 degree
rightwards double arrow angle BDA space equals space 180 degree space minus space 90 degree space minus space 50 degree equals space 40 degree
Consider space triangle ABD space & space triangle BAC
AD space equals space BC comma space angle DAB space equals space angle ABC space equals space 90 degree comma space BD space equals space AC
Hence comma space by space RHS space property space triangle ABD space approximately equal to space triangle BAC
rightwards double arrow angle ABD space equals space angle BAC space equals space 50 degree
Now comma space consider space triangle ABP
angle PAB space plus space angle PBA space plus space angle APB space equals 180 degree
angle PAB space equals space angle BAC space equals space 50 degree
angle PBA space equals space angle ABD space equals space 50 degree
rightwards double arrow angle APB space equals space 180 degree space minus space 50 degree space minus space 50 degree space equals 80 degree
Now comma space angle APB space equals space angle DPC space left parenthesis Opposite space angles right parenthesis
rightwards double arrow angle DPC space equals space 80 degree
Hence comma space correct space option space is space left parenthesis straight c right parenthesis. end style

CBSE Class 9 Maths Homework Help

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