NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals

NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals

Edited By Ramraj Saini | Updated on Sep 27, 2023 10:40 PM IST

NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals

Quadrilaterals class 9 questions and answers are discussed here. These NCERT solutions are developed by the expert at Careers360 taking care of latest CBSE syllabus 2023. the solutions are provided for all the NCERT problems in simple and easy to understand language covering step by step all the concepts which ultimately help during the exam.

In class 9 maths chapter 8 question answer, you will get to know about angles sum property, rhombus, parallelogram, square, rectangle, trapezium, kite, etc. The chapter is starting with recollecting the angle sum property of a quadrilateral. In total there are 2 exercises that consist of 37 questions. NCERT solutions for class 9 maths chapter 8 Quadrilaterals has the solutions to all the 37 questions. Here you will get NCERT solutions for class 9 Maths also.

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Quadrilaterals Class 9 Solutions - Important Formulae And Points

Quadrilateral:

  • Sum of all angles = 360°

Parallelogram:

  • A diagonal of a parallelogram divides it into two congruent triangles.

  • In a parallelogram, diagonals bisect each other.

  • In a parallelogram, opposite angles are equal.

  • In a parallelogram, opposite sides are equal.

Square:

  • Diagonals of a square bisect each other at right angles and are equal, and vice-versa.

Triangle:

  • A line through the midpoint of a side of a triangle parallel to another side bisects the third side (Midpoint theorem).

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

Parallelogram Angle Bisectors:

  • In a parallelogram, the bisectors of any two consecutive angles intersect at a right angle.

  • If a diagonal of a parallelogram bisects one of the angles of a parallelogram, it also bisects the second angle.

Rectangle:

  • The angle bisectors of a parallelogram form a rectangle.

  • Each of the four angles of a rectangle is a right angle.

Rhombus:

  • The diagonals of a rhombus are perpendicular to each other.

Free download NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals for CBSE Exam.

Quadrilaterals Class 9 NCERT Solutions (Intext Questions and Exercise)

Class 9 maths chapter 8 NCERT solutions - Exercise: 8.1

Q1 The angles of quadrilateral are in the ratio \small 3:5:9:13 . Find all the angles of the quadrilateral.

Answer:

Given : The angles of a quadrilateral are in the ratio \small 3:5:9:13 .
Let the angles of quadrilateral be 3x,5x,9x ,13x .

Sum of all angles is 360.

Thus, 3x+5x+9x +13x=360 \degree

\Rightarrow 30x=360 \degree

\Rightarrow x=12 \degree

All four angles are : 3x=3\times 12=36\degree

5x=5\times 12=60 \degree

9x=9\times 12=108 \degree

13x=13\times 12=156 \degree

Q2 If the diagonals of a parallelogram are equal, then show that it is a rectangle.

Answer:

Given: ABCD is a parallelogram with AC=BD.

To prove: ABCD is a rectangle.

Proof : In \triangle ABC and \triangle BAD,

1640170405939

BC= AD (Opposite sides of parallelogram)

AC=BD (Given)

AB=AB (common)

\triangle ABC \cong \triangle BAD (By SSS)

\angle ABC=\angle BAD (CPCT)

and \angle ABC+\angle BAD=180 \degree (co - interior angles)

2\angle BAD=180 \degree

\angle BAD= 90 \degree

Hence, it is a rectangle.

Q3 Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.

Answer:

Given: the diagonals of a quadrilateral bisect each other at right angles i.e.BO=OD.

To prove: ABCD is a rhombus.

Proof : In \triangle AOB and \triangle AOD,

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\angle AOB=\angle AOD (Each 90 \degree )

BO=OD (Given )

AO=AO (common)

\triangle AOB \cong \triangle AOD (By SAS)

AB=AD (CPCT)

Similarly, AB=BC and BC=CD

Hence, it is a rhombus.

Q4 Show that the diagonals of a square are equal and bisect each other at right angles.

Answer:

Given : ABCD is a square i.e. AB=BC=CD=DA.

To prove : the diagonals of a square are equal and bisect each other at right angles i.e. AC=BD,AO=CO,BO=DO and \angle COD=90 \degree

Proof : In \triangle BAD and \triangle ABC,

1651648000412

\angle BAD = \angle ABC (Each 90 \degree )

AD=BC (Given )

AB=AB (common)

\triangle BAD \cong \triangle ABC (By SAS)

BD=AC (CPCT)

In \triangle AOB and \triangle COD,

\angle OAB= \angle OCD (Alternate angles)

AB=CD (Given )

\angle OBA= \angle ODC (Alternate angles)

\triangle AOB \cong \triangle COD (By AAS)

AO=OC ,BO=OD (CPCT)

In \triangle AOB and \triangle AOD,

OB=OD (proved above)

AB=AD (Given )

OA=OA (COMMON)

\triangle AOB \cong \triangle AOD (By SSS)

\angle AOB= \angle AOD (CPCT)

\angle AOB+ \angle AOD = 180 \degree

2. \angle AOB = 180 \degree

\angle AOB = 90 \degree

Hence, the diagonals of a square are equal and bisect each other at right angles.

Q5 Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.

Answer:

Given : ABCD is a quadrilateral with AC=BD,AO=CO,BO=DO, \angle COD = 90 \degree

To prove: ABCD is a square.

Proof: Since the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a rhombus.

Thus, AB=BC=CD=DA

In \triangle BAD and \triangle ABC,

AD=BC (proved above )

AB=AB (common)

BD=AC

\triangle BAD \cong \triangle ABC (By SSS)

\angle BAD = \angle ABC (CPCT)

\angle BAD+ \angle ABC = 180 \degree (Co-interior angles)

2. \angle ABC = 180 \degree

\angle ABC = 90 \degree

Hence, the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.

Q6 (i) Diagonal AC of a parallelogram ABCD bisects \small \angle A (see Fig. \small 8.19 ). Show that

it bisects \small \angle C also.

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Answer:

Given: \angle DAC= \angle BAC ................1

\angle DAC= \angle BCA.................2 (Alternate angles)

\angle BAC= \angle ACD .................3 (Alternate angles)

From equation 1,2 and 3, we get

\angle ACD= \angle BCA...................4

Hence, diagonal AC bisect angle C also.

Q6 (ii) Diagonal AC of a parallelogram ABCD bisects \small \angle A (see Fig. \small 8.19 ). Show that

ABCD is a rhombus.

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Answer:

Given: \angle DAC= \angle BAC ................1

\angle DAC= \angle BCA.................2 (Alternate angles)

\angle BAC= \angle ACD .................3 (Alternate angles)

From equation 1,2 and 3, we get

\angle ACD= \angle BCA...................4

From 2 and 4, we get

\angle ACD= \angle DAC

In \triangle ADC,

\angle ACD= \angle DAC (proved above )

AD=DC (In a triangle,sides opposite to equal angle are equal)

A parallelogram whose adjacent sides are equal , is a rhombus.

Thus, ABCD is a rhombus.

Q7 ABCD is a rhombus. Show that diagonal AC bisects \small \angle A as well as \small \angle C and diagonal BD bisects \small \angle B as well as \small \angle D .

Answer:

1640170558677

In \triangle ADC,

AD = CD (ABCD is a rhombus)

\angle 3= \angle 1.................1(angles opposite to equal sides are equal )

\angle 3= \angle 2.................2 (alternate angles)

From 1 and 2, we have

\angle 1= \angle 2.................3

and \angle 1= \angle 4.................4 (alternate angles)

From 1 and 4, we get

\angle 3= \angle 4.................5

Hence, from 3 and 5, diagonal AC bisects angles A as well as angle C.

1640170587935

In \triangle ADB,

AD = AB (ABCD is a rhombus)

\angle 5= \angle 7.................6(angles opposite to equal sides are equal )

\angle 7= \angle 6.................7 (alternate angles)

From 6 and 7, we have

\angle 5= \angle 6.................8

and \angle 5= \angle 8.................9(alternate angles)

From 6 and 9, we get

\angle 7= \angle 8.................10

Hence, from 8 and 10, diagonal BD bisects angles B as well as angle D.

Q8 (i) ABCD is a rectangle in which diagonal AC bisects \small \angle A as well as \small \angle C . Show that:

ABCD is a square

Answer:

1640170617153

Given: ABCD is a rectangle with AB=CD and BC=AD \angle 1= \angle 2 and \angle 3= \angle 4.

To prove: ABCD is a square.

Proof : \angle 1= \angle 4 .............1(alternate angles)

\angle 3= \angle 4 ................2(given )

From 1 and 2, \angle 1= \angle 3.....................................3

In \triangle ADC,

\angle 1= \angle 3 (from 3 )

DC=AD (In a triangle, sides opposite to equal angle are equal )

A rectangle whose adjacent sides are equal is a square.

Hence, ABCD is a square.

Q8 (ii) ABCD is a rectangle in which diagonal AC bisects \small \angle A as well as \small \angle C . Show that:

diagonal BD bisects \small \angle B as well as \small \angle D .

Answer: 1651648175820

In \triangle ADB,

AD = AB (ABCD is a square)

\angle 5= \angle 7.................1(angles opposite to equal sides are equal )

\angle 5= \angle 8.................2 (alternate angles)

From 1 and 2, we have

\angle 7= \angle 8.................3

and \angle 7 = \angle 6.................4(alternate angles)

From 1 and 4, we get

\angle 5= \angle 6.................5

Hence, from 3 and 5, diagonal BD bisects angles B as well as angle D.

Q9 (i) In parallelogram ABCD, two points P and Q are taken on diagonal BD such that \small DP=BQ (see Fig. \small 8.20 ). Show that: \small \Delta APD\cong \Delta CQB

1640170652426

Answer:

Given: In parallelogram ABCD, two points P and Q are taken on diagonal BD such that \small DP=BQ .

To prove : \small \Delta APD\cong \Delta CQB

Proof :

In \small \Delta APD\, \, and\, \, \Delta CQB,

DP=BQ (Given )

\angle ADP= \angle CBQ (alternate angles)

AD=BC (opposite sides of a parallelogram)

\small \Delta APD\cong \Delta CQB (By SAS)

Q9 (ii) In parallelogram ABCD, two points P and Q are taken on diagonal BD such that \small DP=BQ (see Fig. \small 8.20 ). Show that: \small AP=CQ

1640170672862

Answer:

Given: In parallelogram ABCD, two points P and Q are taken on diagonal BD such that \small DP=BQ .

To prove : \small AP=CQ

Proof :

In \small \Delta APD\, \, and\, \, \Delta CQB,

DP=BQ (Given )

\angle ADP= \angle CBQ (alternate angles)

AD=BC (opposite sides of a parallelogram)

\small \Delta APD\cong \Delta CQB (By SAS)

\small AP=CQ (CPCT)

Q9 (iii) In parallelogram ABCD, two points P and Q are taken on diagonal BD such that \small DP=BQ (see Fig. \small 8.20 ). Show that: \small \Delta AQB\cong \Delta CPD

1640170699729

Answer:

Given: In parallelogram ABCD, two points P and Q are taken on diagonal BD such that \small DP=BQ .

To prove : \small \Delta AQB\cong \Delta CPD

Proof :

In \small \Delta AQB\, \, and\, \, \Delta CPD,

DP=BQ (Given )

\angle ABQ= \angle CDP (alternate angles)

AB=CD (opposite sides of a parallelogram)

\small \Delta AQB\cong \Delta CPD (By SAS)

Q9 (iv) In parallelogram ABCD, two points P and Q are taken on diagonal BD such that \small DP=BQ (see Fig. \small 8.20 ). Show that: \small AQ=CP

1640170719974

Answer:

Given: In parallelogram ABCD, two points P and Q are taken on diagonal BD such that \small DP=BQ .

To prove : \small AQ=CP

Proof :

In \small \Delta AQB\, \, and\, \, \Delta CPD,

DP=BQ (Given )

\angle ABQ= \angle CDP (alternate angles)

AB=CD (opposite sides of a parallelogram)

\small \Delta AQB\cong \Delta CPD (By SAS)

\small AQ=CP (CPCT)

Q9 (v) In parallelogram ABCD, two points P and Q are taken on diagonal BD such that \small DP=BQ (see Fig. \small 8.20 ). Show that: APCQ is a parallelogram

1640170738363

Answer:

Given: In parallelogram ABCD, two points P and Q are taken on diagonal BD such that \small DP=BQ .

To prove: APCQ is a parallelogram

Proof :

In \small \Delta APD\, \, and\, \, \Delta CQB,

DP=BQ (Given )

\angle ADP= \angle CBQ (alternate angles)

AD=BC (opposite sides of a parallelogram)

\small \Delta APD\cong \Delta CQB (By SAS)

\small AP=CQ (CPCT)...............................................................1

Also,

In \small \Delta AQB\, \, and\, \, \Delta CPD,

DP=BQ (Given )

\angle ABQ= \angle CDP (alternate angles)

AB=CD (opposite sides of a parallelogram)

\small \Delta AQB\cong \Delta CPD (By SAS)

\small AQ=CP (CPCT)........................................2

From equation 1 and 2, we get

\small AP=CQ

\small AQ=CP

Thus, opposite sides of quadrilateral APCQ are equal so APCQ is a parallelogram.

Q10 (i) ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD (see Fig. \small 8.21 ). Show that \small \Delta APB\cong \Delta CQD

1640170770148

Answer:

Given: ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD.

To prove : \small \Delta APB\cong \Delta CQD

Proof: In \small \Delta APB\, \, and\, \, \Delta CQD ,

\angle APB= \angle CQD (Each 90 \degree )

\angle ABP= \angle CDQ (Alternate angles)

AB=CD (Opposite sides of a parallelogram )

Thus, \small \Delta APB\cong \Delta CQD (By SAS)

Q10 (ii) ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD (see Fig. \small 8.21 ). Show that \small AP=CQ

1640170798432

Answer:

Given: ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD.

To prove : \small AP=CQ

Proof: In \small \Delta APB\, \, and\, \, \Delta CQD ,

\angle APB= \angle CQD (Each 90 \degree )

\angle ABP= \angle CDQ (Alternate angles)

AB=CD (Opposite sides of a parallelogram )

Thus, \small \Delta APB\cong \Delta CQD (By SAS)

\small AP=CQ (CPCT)

Q11 (i) In \small \Delta ABC and \small \Delta DEF , \small AB=DE,AB\parallel DE,BC=EF and \small BC\parallel EF . Vertices A, B and C are joined to vertices D, E and F respectively (see Fig. \small 8.22 ). Show that quadrilateral ABED is a parallelogram

1640170849147

Answer:

Given : In \small \Delta ABC and \small \Delta DEF , \small AB=DE,AB\parallel DE,BC=EF and \small BC\parallel EF .

To prove : quadrilateral ABED is a parallelogram

Proof : In ABED,

AB=DE (Given)

AB||DE (Given )

Hence, quadrilateral ABED is a parallelogram.

Q11 (ii) In \small \Delta ABC and \small \Delta DEF , \small AB=DE,AB\parallel DE,BC=EF and \small BC\parallel EF . Vertices A, B and C are joined to vertices D, E and F respectively (see Fig. \small 8.22 ). Show that quadrilateral BEFC is a parallelogram.

1640170880705

Answer:

Given: In \small \Delta ABC and \small \Delta DEF , \small AB=DE,AB\parallel DE,BC=EF and \small BC\parallel EF .

To prove: quadrilateral BEFC is a parallelogram

Proof: In BEFC,

BC=EF (Given)

BC||EF (Given )

Hence, quadrilateral BEFC is a parallelogram.

Q11 (iii) In \small \Delta ABC and \small \Delta DEF , \small AB=DE,AB\parallel DE,BC=EF and \small BC\parallel EF . Vertices A, B and C are joined to vertices D, E and F respectively (see Fig. \small 8.22 ). Show that \small AD\parallel CF and \small AD=CF

1640170909196

Answer:

To prove: \small AD\parallel CF and \small AD=CF

Proof :

In ABED,

AD=BE.................1(ABED is a parallelogram)

AD||BE .................2(ABED is a parallelogram)

In BEFC,

BE=CF.................3(BEFC is a parallelogram)

BE||CF .................4(BEFC is a parallelogram)

From 2 and 4 , we get

AD||CF

From 1 and 3, we get

AD=CF

Q11 (iv) In \small \Delta ABC and \small \Delta DEF , \small AB=DE,AB\parallel DE,BC=EF and \small BC\parallel EF. . Vertices A, B and C are joined to vertices D, E and F respectively (see Fig. \small 8.22 ). Show that quadrilateral ACFD is a parallelogram.

1640170928287

Answer:

To prove : quadrilateral ACFD is a parallelogram

Proof :

In ABED,

AD=BE.................1(ABED is a parallelogram)

AD||BE .................2(ABED is a parallelogram)

In BEFC,

BE=CF.................3(BEFC is a parallelogram)

BE||CF .................4(BEFC is a parallelogram)

From 2 and 4 , we get

AD||CF...........................5

From 1 and 3, we get

AD=CF...........................6

From 5 and 6, we get

AD||CF and AD=CF

Thus, quadrilateral ACFD is a parallelogram

Q11 (v) In \small \Delta ABC and \small \Delta DEF , \small AB=DE,AB\parallel DE,BC = EF and \small BC\parallel EF . Vertices A, B and C are joined to vertices D, E and F respectively (see Fig. \small 8.22 ). Show that \small AC=DF

1640170956375

Answer:

In ACFD,

AC=DF (Since, ACFD is a parallelogram which is prooved in part (iv) of the question)

Q11 (vi) In \small \Delta ABC and \small \Delta DEF , \small AB=DE,AB\parallel DE,BC=EF and \small BC\parallel EF. Vertices A, B and C are joined to vertices D, E and F respectively (see Fig. \small 8.22 ). Show that \small \Delta ABC\cong \Delta DEF

1640170982314

Answer:

In \small \Delta ABC and \small \Delta DEF ,

AB=DE (Given )

BC=EF (Given )

AC=DF ( proved in (v) part)

\small \Delta ABC\cong \Delta DEF (By SSS rule)

Q12 (i) ABCD is a trapezium in which \small AB\parallel CD and \small AD=BC (see Fig. \small 8.23 ). Show that \small \angle A=\angle B

[ Hint: Extend AB and draw a line through C parallel to DA intersecting AB produced at E.]

1640171017353

Answer:

Given: ABCD is a trapezium in which \small AB\parallel CD and \small AD=BC

To prove : \small \angle A=\angle B

Proof: Let \angle A be \angle 1, \angle ABC be \angle 2, \angle EBC be \angle 3, \angle BEC be \angle 4.

In AECD,

AE||DC (Given)

AD||CE (By cnstruction)

Hence, AECD is a parallelogram.

AD=CE...............1(opposite sides of a parallelogram)

AD=BC.................2(Given)

From 1 and 2, we get

CE=BC

In \triangle BCE,

\angle 3=\angle 4 .................3 (opposite angles of equal sides)

\angle 2+\angle 3=180 \degree ...................4(linear pairs)

\angle 1+\angle 4=180 \degree .....................5(Co-interior angles)

From 4 and 5, we get

\angle 2+\angle 3=\angle 1+\angle 4

\therefore \angle 2=\angle 1 \rightarrow \angle B=\angle A (Since, \angle 3=\angle 4 )

Q12 (ii) ABCD is a trapezium in which AB || CD and AD = BC (see Fig. 8.23). Show that \small \angle C=\angle D

[ Hint : Extend AB and draw a line through C parallel to DA intersecting AB produced at E.]

1640171044691

Answer:

Given: ABCD is a trapezium in which \small AB\parallel CD and \small AD=BC

To prove : \small \angle C=\angle D

Proof: Let \angle A be \angle 1, \angle ABC be \angle 2, \angle EBC be \angle 3, \angle BEC be \angle 4.

\angle 1+\angle D= 180 \degree (Co-interior angles)

\angle 2+\angle C= 180 \degree (Co-interior angles)

\therefore \angle 1+\angle D=\angle 2+\angle C

Thus, \small \angle C=\angle D (Since , \small \angle 1=\angle 2 )

Q12 (iii) ABCD is a trapezium in which \small AB\parallel CD and \small AD=BC (see Fig. \small 8.23 ). Show that \small \Delta ABC\cong \Delta BAD

[ Hint : Extend AB and draw a line through C parallel to DA intersecting AB produced at E.]

1640171072699

Answer:

Given: ABCD is a trapezium in which \small AB\parallel CD and \small AD=BC

To prove : \small \Delta ABC\cong \Delta BAD

Proof: In \small \Delta ABC\, \, and\, \, \, \Delta BAD ,

BC=AD (Given )

AB=AB (Common )

\angle ABC=\angle BAD (proved in (i) )

Thus, \small \Delta ABC\cong \Delta BAD (By SAS rule)

Q12 (iv) ABCD is a trapezium in which \small AB\parallel CD and \small AD=BC (see Fig. \small 8.23 ). Show that diagonal AC \small = diagonal BD

[ Hint : Extend AB and draw a line through C parallel to DA intersecting AB produced at E.]

1640171092180

Answer:

Given: ABCD is a trapezium in which \small AB\parallel CD and \small AD=BC

To prove: diagonal AC \small = diagonal BD

Proof: In \small \Delta ABC\, \, and\, \, \, \Delta BAD ,

BC=AD (Given )

AB=AB (Common )

\angle ABC=\angle BAD (proved in (i) )

Thus, \small \Delta ABC\cong \Delta BAD (By SAS rule)

diagonal AC \small = diagonal BD (CPCT)

Class 9 maths chapter 8 question answer - Exercise : 8.2

Q1 (i) ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA (see Fig \small 8.29 ). AC is a diagonal. Show that : \small SR\parallel AC and \small SR=\frac{1}{2}AC

1640171132022

Answer:

Given: ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA (see Fig \small 8.29 ). AC is a diagonal.

To prove :

\small SR\parallel AC and \small SR=\frac{1}{2}AC

Proof: In \triangle ACD,

S is the midpoint of DA. (Given)

R is the midpoint of DC. (Given)

By midpoint theorem,

\small SR\parallel AC and \small SR=\frac{1}{2}AC

Q1 (ii) ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA (see Fig \small 8.29 ). AC is a diagonal. Show that : \small PQ=SR

1640171177941

Answer:

Given: ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA (see Fig \small 8.29 ). AC is a diagonal.

To prove : \small PQ=SR

Proof : In \triangle ACD,

S is mid point of DA. (Given)

R is mid point of DC. (Given)

By mid point theorem,

\small SR\parallel AC and \small SR=\frac{1}{2}AC ...................................1

In \triangle ABC,

P is mid point of AB. (Given)

Q is mid point of BC. (Given)

By mid point theorem,

\small PQ\parallel AC and \small PQ=\frac{1}{2}AC .................................2

From 1 and 2,we get

\small PQ\parallel SR and \small PQ=SR=\frac{1}{2}AC

Thus, \small PQ=SR

Q1 (iii) ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA (see Fig \small 8.29 ). AC is a diagonal. Show that : PQRS is a parallelogram.

1640171205589

Answer:

Given : ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA (see Fig \small 8.29 ). AC is a diagonal.

To prove : PQRS is a parallelogram.

Proof : In PQRS,

Since,

\small PQ\parallel SR and \small PQ=SR .

So,PQRS is a parallelogram.

Q2 ABCD is a rhombus and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rectangle.

Answer:

1640171236054

Given: ABCD is a rhombus in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA . AC, BD are diagonals.

To prove: the quadrilateral PQRS is a rectangle.

Proof: In \triangle ACD,

S is midpoint of DA. (Given)

R is midpoint of DC. (Given)

By midpoint theorem,

\small SR\parallel AC and \small SR=\frac{1}{2}AC ...................................1

In \triangle ABC,

P is midpoint of AB. (Given)

Q is mid point of BC. (Given)

By mid point theorem,

\small PQ\parallel AC and \small PQ=\frac{1}{2}AC .................................2

From 1 and 2,we get

\small PQ\parallel SR and \small PQ=SR=\frac{1}{2}AC

Thus, \small PQ=SR and \small PQ\parallel SR

So,the quadrilateral PQRS is a parallelogram.

Similarly, in \triangle BCD,

Q is mid point of BC. (Given)

R is mid point of DC. (Given)

By mid point theorem,

\small QR\parallel BD

So, QN || LM ...........5

LQ || MN ..........6 (Since, PQ || AC)

From 5 and 6, we get

LMPQ is a parallelogram.

Hence, \small \angle LMN= \small \angle LQN (opposite angles of the parallelogram)

But, \small \angle LMN= 90 (Diagonals of a rhombus are perpendicular)

so, \small \angle LQN=90

Thus, a parallelogram whose one angle is right angle,ia a rectangle.Hence,PQRS is a rectangle.

Q3 ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rhombus.

Answer:

1640172120681

Given: ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA respectively.

To prove: the quadrilateral PQRS is a rhombus.

Proof :

In \triangle ACD,

S is the midpoint of DA. (Given)

R is the midpoint of DC. (Given)

By midpoint theorem,

\small SR\parallel AC and \small SR=\frac{1}{2}AC ...................................1

In \triangle ABC,

P is the midpoint of AB. (Given)

Q is the midpoint of BC. (Given)

By midpoint theorem,

\small PQ\parallel AC and \small PQ=\frac{1}{2}AC .................................2

From 1 and 2, we get

\small PQ\parallel SR and \small PQ=SR=\frac{1}{2}AC

Thus, \small PQ=SR and \small PQ\parallel SR

So, the quadrilateral PQRS is a parallelogram.

Similarly, in \triangle BCD,

Q is the midpoint of BC. (Given)

R is the midpoint of DC. (Given)

By midpoint theorem,

\small QR\parallel BD and \small QR=\frac{1}{2}BD ...................5

AC = BD.......................6(diagonals )

From 2, 5 and 6, we get

PQ=QR

Thus, a parallelogram whose adjacent sides are equal is a rhombus. Hence, PQRS is a rhombus.

Q4 ABCD is a trapezium in which \small AB\parallel DC , BD is a diagonal and E is the mid-point of AD. A line is drawn through E parallel to AB intersecting BC at F (see Fig. \small 8.30 ). Show that F is the mid-point of BC.

1640172147378

Answer:

Given: ABCD is a trapezium in which \small AB\parallel DC , BD is a diagonal and E is the mid-point of AD. A line is drawn through E parallel to AB intersecting BC at F (see Fig. \small 8.30 ).

To prove: F is the mid-point of BC.

In \triangle ABD,

E is the midpoint of AD. (Given)

EG || AB (Given)

By converse of midpoint theorem,

G is the midpoint of BD.

In \triangle BCD,

G is mid point of BD. (Proved above)

FG || DC (Given)

By converse of midpoint theorem,

F is the midpoint of BC.

Q5 In a parallelogram ABCD, E and F are the mid-points of sides AB and CD respectively (see Fig. \small 8.31 ). Show that the line segments AF and EC trisect the diagonal BD.

1640172188917

Answer:

Given: In a parallelogram ABCD, E and F are the mid-points of sides AB and CD respectively

To prove: the line segments AF and EC trisect the diagonal BD.

Proof : In quadrilatweral ABCD,

AB=CD (Given)

\frac{1}{2}AB=\frac{1}{2}CD

\Rightarrow AE=CF (E and F are midpoints of AB and CD)

In quadrilateral AECF,

AE=CF (Given)

AE || CF (Opposite sides of a parallelogram)

Hence, AECF is a parallelogram.

In \triangle DCQ,

F is the midpoint of DC. (given )

FP || CQ (AECF is a parallelogram)

By converse of midpoint theorem,

P is the mid point of DQ.

DP= PQ....................1

Similarly,

In \triangle ABP,

E is the midpoint of AB. (given )

EQ || AP (AECF is a parallelogram)

By converse of midpoint theorem,

Q is the midpoint of PB.

OQ= QB....................2

From 1 and 2, we have

DP = PQ = QB.

Hence, the line segments AF and EC trisect the diagonal BD.

Q6 Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.

Answer:

1640172226300

Given: ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA. AC, BD are diagonals.

To prove: the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.

Proof: In \triangle ACD,

S is the midpoint of DA. (Given)

R is midpoint of DC. (Given)

By midpoint theorem,

\small SR\parallel AC and \small SR=\frac{1}{2}AC ...................................1

In \triangle ABC,

P is the midpoint of AB. (Given)

Q is the midpoint of BC. (Given)

By midpoint theorem,

\small PQ\parallel AC and \small PQ=\frac{1}{2}AC .................................2

From 1 and 2, we get

\small PQ\parallel SR and \small PQ=SR=\frac{1}{2}AC

Thus, \small PQ=SR and \small PQ\parallel SR

So, the quadrilateral PQRS is a parallelogram and diagonals of a parallelogram bisect each other.

Thus, SQ and PR bisect each other.

Q7 (i) ABC is a triangle right angled at C. A line through the mid-point M of hypotenuse AB and parallel to BC intersects AC at D. Show that D is the mid-point of AC.

Answer:

1640172246282

Given: ABC is a triangle right angled at C. A line through the mid-point M of hypotenuse AB and parallel to BC intersects AC at D.

To prove :D is mid point of AC.

Proof: In \triangle ABC,

M is mid point of AB. (Given)

DM || BC (Given)

By converse of mid point theorem,

D is the mid point of AC.

Q7 (ii) ABC is a triangle right angled at C. A line through the mid-point M of hypotenuse AB and parallel to BC intersects AC at D. Show that \small MD\perp AC

Answer:

1651648435406

Given: ABC is a triangle right angled at C. A line through the mid-point M of hypotenuse AB and parallels to BC intersects AC at D.

To prove : \small MD\perp AC

Proof : \angle ADM = \angle ACB (Corresponding angles)

\angle ADM= 90 \degree . ( \angle ACB = 90 \degree )

Hence, \small MD\perp AC .

Q7 (iii) ABC is a triangle right angled at C. A line through the mid-point M of hypotenuse AB and parallel to BC intersects AC at D. Show that \small CM=MA=\frac{1}{2}AB

Answer:

1651648482141

Given: ABC is a triangle right angled at C. A line through the mid-point M of hypotenuse AB and parallel to BC intersects AC at D.

To prove : \small CM=MA=\frac{1}{2}AB

Proof : In \triangle ABC,

M is the midpoint of AB. (Given)

DM || BC (Given)

By converse of midpoint theorem,

D is the midpoint of AC i.e. AD = DC.

In \triangle AMD and \triangle CMD,

AD = DC (proved above)

\angle ADM = \angle CDM (Each right angle)

DM = DM (Common)

\triangle AMD \cong \triangle CMD (By SAS)

AM = CM (CPCT)

But , \small AM=\frac{1}{2}AB

Hence, \small CM=MA=\frac{1}{2}AB .

Summary Of Class 9 Quadrilaterals NCERT Solutions

Following topics are covered in the chapter:

  • Angle Sum Property
  • Types of Quadrilaterals
  • Properties of a Parallelogram
  • Condition for a Quadrilateral to be a Parallelogram
  • The Mid-point Theorem

Important Concepts Learned in NCERT Maths Chapter 8 Class 9 – Quadrilaterals

  • A quadrilateral is a polygon with four sides and four angles.
  • The sum of the angles in a quadrilateral is 360 degrees.
  • The opposite sides of a parallelogram are parallel and congruent.
  • The opposite angles of a parallelogram are congruent.
  • The diagonals of a parallelogram bisect each other.
  • A rectangle is a parallelogram with four right angles.
  • The diagonals of a rectangle are congruent and bisect each other.
  • A square is a rectangle with four congruent sides.
  • The diagonals of a square are congruent and bisect each other at right angles.
  • A rhombus is a parallelogram with four congruent sides.
  • The diagonals of a rhombus bisect each other at right angles.
  • The area of a quadrilateral can be calculated by dividing it into triangles and using the formula for the area of a triangle.
  • The perimeter of a quadrilateral is the sum of the lengths of its sides.
  • The properties of quadrilaterals can be used to solve problems related to real-life situations.

Quadrilaterals Class 9 Solutions - Exercise Wise

Students can practice class 9 maths ch 8 question answer using the exercise line given below.

NCERT Solutions For Class 9 Maths - Chapter Wise

Chapter No.
Chapter Name
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Quadrilaterals
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15

NCERT Solutions For Class 9 - Subject Wise

How To Use NCERT Solutions For Class 9 Maths Chapter 8 Quadrilaterals

  • Read and memorize the properties related to all kinds of quadrilaterals.
  • Have a glance through some solved to understand the pattern of the solution to that particular question.
  • Now check your learning on the practice exercises problems.
  • If you stuck in any question then you can assist yourself using NCERT solutions for class 9 maths chapter 8 Quadrilaterals.
  • Following the above-written steps, you can get 100% out of the chapter.

Also Check NCERT Books and NCERT Syllabus here:

Frequently Asked Questions (FAQs)

1. What are the important topics in Quadrilaterals ch 8 maths class 9?

Angle sum property of a quadrilateral, properties of the parallelogram, another condition for a quadrilateral to be a parallelogram, and mid-point theorem are the important topics covered in class 9th quadrilateral solution. students can practice these NCERT solutions for class 9 to command the concepts.

2. How does the NCERT solutions are helpful ?

NCERT solutions are not only helpful for the students if they stuck while solving NCERT problems but also it will provide them new ways to solve the problems. These solutions are provided in a very detailed manner which will give them conceptual clarity.

3. In NCERT Solutions for Class 9 Maths Chapter 8, how does the concept of "quadrilaterals" get defined?

Quadrilateral class 9 solutions define a quadrilateral as a two-dimensional shape that has four sides or edges and four corners or vertices. Quadrilaterals are commonly recognized by their standard shapes, such as rectangle, square, trapezoid, and kite, but they can also have irregular and undefined shapes.

4. Where can I find the complete solutions of class 9 chapter 8 maths ?

Here you will get the detailed NCERT solutions for class 9  by clicking on the link. for ease students can study quadrilateral class 9 pdf both online and offline mode. 

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A block of mass 0.50 kg is moving with a speed of 2.00 ms-1 on a smooth surface. It strikes another mass of 1.00 kg and then they move together as a single body. The energy loss during the collision is

Option 1)

0.34\; J

Option 2)

0.16\; J

Option 3)

1.00\; J

Option 4)

0.67\; J

A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1 m 1000 times.  Assume that the potential energy lost each time he lowers the mass is dissipated.  How much fat will he use up considering the work done only when the weight is lifted up ?  Fat supplies 3.8×107 J of energy per kg which is converted to mechanical energy with a 20% efficiency rate.  Take g = 9.8 ms−2 :

Option 1)

2.45×10−3 kg

Option 2)

 6.45×10−3 kg

Option 3)

 9.89×10−3 kg

Option 4)

12.89×10−3 kg

 

An athlete in the olympic games covers a distance of 100 m in 10 s. His kinetic energy can be estimated to be in the range

Option 1)

2,000 \; J - 5,000\; J

Option 2)

200 \, \, J - 500 \, \, J

Option 3)

2\times 10^{5}J-3\times 10^{5}J

Option 4)

20,000 \, \, J - 50,000 \, \, J

A particle is projected at 600   to the horizontal with a kinetic energy K. The kinetic energy at the highest point

Option 1)

K/2\,

Option 2)

\; K\;

Option 3)

zero\;

Option 4)

K/4

In the reaction,

2Al_{(s)}+6HCL_{(aq)}\rightarrow 2Al^{3+}\, _{(aq)}+6Cl^{-}\, _{(aq)}+3H_{2(g)}

Option 1)

11.2\, L\, H_{2(g)}  at STP  is produced for every mole HCL_{(aq)}  consumed

Option 2)

6L\, HCl_{(aq)}  is consumed for ever 3L\, H_{2(g)}      produced

Option 3)

33.6 L\, H_{2(g)} is produced regardless of temperature and pressure for every mole Al that reacts

Option 4)

67.2\, L\, H_{2(g)} at STP is produced for every mole Al that reacts .

How many moles of magnesium phosphate, Mg_{3}(PO_{4})_{2} will contain 0.25 mole of oxygen atoms?

Option 1)

0.02

Option 2)

3.125 × 10-2

Option 3)

1.25 × 10-2

Option 4)

2.5 × 10-2

If we consider that 1/6, in place of 1/12, mass of carbon atom is taken to be the relative atomic mass unit, the mass of one mole of a substance will

Option 1)

decrease twice

Option 2)

increase two fold

Option 3)

remain unchanged

Option 4)

be a function of the molecular mass of the substance.

With increase of temperature, which of these changes?

Option 1)

Molality

Option 2)

Weight fraction of solute

Option 3)

Fraction of solute present in water

Option 4)

Mole fraction.

Number of atoms in 558.5 gram Fe (at. wt.of Fe = 55.85 g mol-1) is

Option 1)

twice that in 60 g carbon

Option 2)

6.023 × 1022

Option 3)

half that in 8 g He

Option 4)

558.5 × 6.023 × 1023

A pulley of radius 2 m is rotated about its axis by a force F = (20t - 5t2) newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10 kg m2 , the number of rotations made by the pulley before its direction of motion if reversed, is

Option 1)

less than 3

Option 2)

more than 3 but less than 6

Option 3)

more than 6 but less than 9

Option 4)

more than 9

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