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

NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals

Answer :

Given: The ratios of the angles of quadrilateral are 3: 5: 9: 13.

Let the angles of the quadrilateral be 3x, 5x, 9x and 13x.

We know that, sum of angles of a quadrilateral =

Thus, we get,

angles of quadrilateral

=> 3x = 3 ×

=>5x = 5 ×

=> 9x = 9 ×

=> 13x = 13 ×

NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals

Answer :

Given: Parallelogram is ABCD whose diagonals AC and BD are equal.

To prove: ABCD is a rectangle.

Proof:

In

AC = BD ...(Given)

AB = CD

(Opposite sides of parallelogram)

BC = CB ...(Common sides)

(By SSS rule)

But from figure, DC || AB and transversal CB intersects them

(interior angles on same side of transversal)

(from(i))

Also,

Thus, we can say that, ABCD is a parallelogram and some angles are

Hence it is proved that, ABCD is a rectangle.

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

NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals

Answer :

Given: A quadrilateral ABCD whose diagonals AC and BD bisect each other at right angles.

And also,

To prove: ABCD is a rhombus.

Proof:

In

OA = OC and OB = OD ...(given)

(vertically opposite angles)

(By SAS rule)

Similarly, in

OA = OC and OD = OB ...(given)

(vertically opposite angles)

Similarly, we can prove that,

AB = CD and

CD = BC ...(iii)

Hence, from (i), (ii) and (iii), we get,

AB = BC = AD = CD

Hence, it is proved that ABCD is a rhombus.

NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals

Answer :

Given: A square ABCD whose diagonals AC and BD intersect at O.

To prove: Diagonals are equal and bisect each other at right angles.

i.e, AC = BD, OD = OB, OA = OC and

Proof: In

BC = AD ...(given)

AB = BA ...(Common side)

Similarly, in

AB = DC ...(given)

(

(

(By SAS rule)

(By CPCT)

Now, in

OA = OD ...(proved earlier)

AB = AD

(sides of square)

AO = OA

(Commom side)

(By SSS rule)

(linear pair axiom)

Also, AC = BD, OA = OC, OB = OD and

Hence, it is proved that diagonals 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.

NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals

Answer :

Given: A quadrilateral ABCD in which AC = BD and AC

To prove: ABCD is a square.

Proof: Let AC and BD intersect at a point O.

In

BO = OD ...(given)

AO = OA ...(Common side)

(By SAS rule)

Also, AB = DC and AD = BC

(opposite sides of parallelogram)

Similarly, in

AC = BD ...(given)

AB = BA ...(Common side)

BC = AD ...(from(i))

(By SSS rule)

But

Hence, it is proved that, ABCD is a square.

Q6 )
Diagonal AC of a parallelogram ABCD bisects

(i) It bisects

(ii) ABCD is a rhombus.

NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals

Answer :

Given, diagonal AC of a parallelogram ABCD bisects

Here, AB || CD and AC is the transversal.

From eq. (i), (ii) and (iii), we get,

Now,

Thus, we can say that, diagonal AC bisects

Now, in

OA = OC

(since, diagonals bisects each other)

DO = OD ...(Common side)

(By SAS rule)

Now, AB = CD and AD = BC

(sides of parallelogram)

Hence it is proved that, ABCD is a rhombus.

Q7 )
ABCD is a rhombus. Show that diagonal AC bisects

NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals

Answer :

Given: ABCD is a rhombus.

i.e., AD = AB = BC = CD ... (i)

To prove: (i) Diagonal AC bisect

(ii) Diagonal BD bisects

Proof: Let AC and BD are the diagonals of rhombus ABCD.

In

AD = AB ...(Given)

AC = CA ...(Common side)

CD = BC ...(From eq. (i))

Thus,

Also,

Also,

And

This shows that, Diagonal AC bisect

Now, in

AB = BC ...(Given)

BD = BD ...(Common side)

AD = CD ...(Given)

Thus,

Also,

Also,

And

This shows that, Diagonal BD bisects

Hence, proved.

Q8 )
ABCD is a rectangle in which diagonal AC bisects

(i) ABCD is a square

(ii) Diagonal BD bisects

NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals

Answer :

Given: ABCD is a rectangle. AB = CD and BC = AD ...(i)

To prove: (i) ABCD is a square. i.e., AB = BC = CD = DA

(ii) Diagonal BD bisects

Proof: In

(Since, AB || DC and AC is transversal that intersects)

Similarly,

AC = CA ...(Common side)

Also, CD = BC ...(ii)

Thus, from eq. (i) and(ii), we get,

AB = BC = AD = CD

Hence, it is proved that ABCD is a square.

Now, in

AB = BC ...(Given)

BO = OB ...(Common side)

OA = OC

(Since,diagonal bisectS each other)

This shows that, Diagonal BD bisects

Similarly, Now, in

AD = CD ...(Given)

OD = DO ...(Common side)

OA = OC

(Since,diagonal bisectS each other)

Hence, it is proved that, Diagonal BD bisects

Q9 )
In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see figure). Show that

i)

ii) AP = CQ

iii)

iv) AQ = CP

v) APCQ is a parallelogram.

NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals

Answer :

Given, ABCD is a parallelogram and P and Q are lie on BD such that DP = BQ ...(i)

i) Now,in

DP = BQ ...(Given)

AD = BC

(Opposite sides are equal in parallelogram)

(Since, AD || BC and BD is a transversal)

ii)

iii) Here, Now, in

DP = BQ ...(Given)

AB = CD

(Opposite sides are equal in parallelogram)

(Since, AB || CD and BD is a transversal)

iv)

v) Now,in

AQ = CP (From part iv))

AP = CQ (From part ii))

PQ = QP ...(Common side)

And

(Vertically Opposite angles)

Now, these equal angles form a pair of alternate angle when line segment AP and QC are intersected by a transversal PQ.

Now, both pairs of opposite sides of quadrilateral APCQ are parallel.

Hence, it is proved that APCQ is a parallelogram.

Q10 )
ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD (see figure). Show that

(i)

(ii) AP = CQ

NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals

Answer :

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

Now, in

CD = AB ...(Sides of parallelogram)

Hence, it is proved.

Q11 )
In

Show that

(i) Quadrilateral ABED is a parallelogram

(ii) Quadrilateral BEFC is a parallelogram

(iii) AD || CF and AD = CF

(iv) Quadrilateral ACFD is a Parallelogram

(v) AC = DF

(vi)

NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals

Answer :

Given: In

(i) Now, in quadrilateral ABED,

AB = DE and AB || DE ...(Given)

Since, a pair of opposite sides is equal and parallel

Therefore, ABED is a parallelogram.

(ii) In quadrilateral BEFC,

BC = EF and BC || EF ...(Given)

Since, a pair of opposite sides is equal and parallel

Therefore, BEFC is a parallelogram.

(iii)Since, ABED is a parallelogram,

AD || BE and AD = BE ...(i)

Also, BEFC in a parallelogram,

CF || BE and CF = BE ...(ii)

Thus, from Eq. (i) and (ii), we get,

AD || CF and AD = CF ...(From part (iii))

Therefore, ACFD is a parallelogram.

(v)Since, ACFD is a parallelogram.

we get, AC = DF and AC || DF

(vi)Now, in

AB = DE ...(Given)

BC = EF ...(Given)

and AC = DF ...(From part (v))

Therefore,

Q12 )
ABCD is a trapezium in which AB || CD and AD = BC (see figure).

Show that

i)

ii)

iii)

iv) Diagonal AC = Diagonal BD

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

NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals

Answer :

Given: ABCD is trapezium.

AB || CD and AD = BC

Now, extend AB and draw a line through C parallel to DA intersecting AB produced at E.

Now, ADCE is a parallelogram.

But AD = BC

i) We know that,

(Since, interior angles on the same side of the transversal )

(Since, BC = EC)

Also,

(Since, ABE is straight line)

Hence, it is proved.

ii)Now,

(Since, interior angles on the same side of the transversal)

(from eq.(i)) ...(ii)

Also,

from Eq. (ii) and (iii), we get,

Hence, proved.

iii)Now, in

AD = BC ...(Given)

AB = BA ...(Common side)

Hence, proved.

iv)

Hence, it is proved.

Q1 )
ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC,
CD and DA (see figure). AC is a diagonal. Show that

i) SR || AC and SR = (1/2) AC

ii) PQ = SR

iii) PQRS is a parallelogram.

NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals

Answer :

Given: P, Q, R, and S are mid-points of the sides.

Therefore, AP = PB, BQ = CQ
CR = DR and AS = DS

i) Now, in

S is the midpoint of AD and R is the mid point of CD.

As, we know that,

By midpoint theorem, the line segment joining the mid points of two sides of a triangle is parallel to the third side.

Thus, we can say that, SR || AC ...(i)

and SR = (

ii) Similarly, now, in

PQ || AC ...(iii)

and PQ = (

Now, from (ii) and (iv), we get,

SR = PQ = (

Now, from (i), (iii) and (v), we get,

PQ || SR and PQ = SR.

(Since, a pair of opposite sides of a quadrilateral PQRS is equal and parallel)

Hence, proved.

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.

NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals

Answer :

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

By midpoint theorm, Now, in

Thus, SR || AC ...(i)

and SR = (

Similarly, in

PQ || AC ...(iii)

and PQ = (

from (i), (ii), (iii) and (iv), we get,

SR = PQ = (

Now, we know that diagonals of a rhombus bisect each other at right angles.

Now, By midpoint theorem, we have,

RQ || BD

Thus, RE || OF

As SR || AC ...(from (i))

Thus, FR || OE

So,

(since, Opposite angle of a quadrilateral is equal)

Thus, PQRS is a parallelogram with

Hence, it is proved that 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.

NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals

Answer :

Given: ABCD is a rectangle.

AB = CD and BC = AD.

Also, P, Q, R and S are mid-points of the sides AB, BC, CD and DA, respectively.

Therefore, by midpoint theorem,

PQ || BD and PQ =

SR || AC and SR =

In rectangle ABCD,

AC = BD