NCERT solution Class 9 quadrilaterals

NCERT Solutions

Class 9 Maths

Quadrilaterals

NCERT Questions

Ex.8.1 Q.1
Ex.8.1 Q.2
Ex.8.1 Q.3
Ex.8.1 Q.4
Ex.8.1 Q.5
Ex.8.1 Q.6
Ex.8.1 Q.7
Ex.8.1 Q.8
Ex.8.1 Q.9
Ex.8.1 Q.10
Ex.8.1 Q.11
Ex.8.1 Q.12
Ex.8.2 Q.1
Ex.8.2 Q.2
Ex.8.2 Q.3
Ex.8.2 Q.4
Ex.8.2 Q.5
Ex.8.2 Q.6
Ex.8.2 Q.7

Ex.8.1 Q.5

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

Let us consider a quadrilateral ABCD in which the diagonals AC and BD intersect each other at O.

It is given that the diagonals of ABCD are equal and bisect each other at right angles.

Therefore, AC = BD, OA = OC, OB = OD, and ∠AOB = ∠BOC = ∠COD = ∠AOD = 90⁰.

To prove ABCD is a square, we have to prove that ABCD is a parallelogram, AB = BC = CD = AD,

and one of its interior angles is 90⁰.

In ΔAOB and ΔCOD,

AO = CO

(Diagonals bisect each other)

OB = OD

(Diagonals bisect each other)

∠AOB = ∠COD

(Vertically opposite angles)

So, ΔAOB ≅ ΔCOD

(SAS congruence rule)

Hence AB = CD………………… (1)

(By CPCT)

And, ∠OAB = ∠OCD

(By CPCT)

However, these are alternate interior angles for line AB and CD and

alternate interior angles are equal to each other only when the two lines are parallel.

So, AB || CD ............(2)

From equations (1) and (2), we obtain

ABCD is a parallelogram.

In ΔAOD and ΔCOD,

AO = CO

(Diagonals bisect each other)

∠AOD = ∠COD

(Given that each is 90⁰)

OD = OD

(Common)

So, ΔAOD ≅ ΔCOD

(SAS congruence rule)

Hence AD = DC ………... (3)

However, AD = BC and AB = CD (Opposite sides of parallelogram ABCD)

=> AB = BC = CD = DA

Therefore, all the sides of quadrilateral ABCD are equal to each other.

In ΔADC and ΔBCD,

AD = BC

(Already proved)

AC = BD

(Given)

DC = CD

(Common)

So, ΔADC ≅ ΔBCD

(SSS Congruence rule)

Hence, ∠ADC = ∠BCD

(By CPCT)

However, ∠ADC + ∠BCD = 180⁰

(Co-interior angles)

=> ∠ADC + ∠ADC = 180⁰

=> 2∠ADC = 180⁰

=> ∠ADC = 90⁰

One of the interior angles of quadrilateral ABCD is a right angle.

Thus, we have obtained that ABCD is a parallelogram, AB = BC = CD = AD and

one of its interior angles is 90⁰. Therefore, ABCD is a square.

Video solution:

Quadrilaterals Class 9 Maths NCERT Chapter 8 NCERT Solutions

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Attempt to Solve Yourself: Before looking at the solution, try to solve or answer the question independently. This practice will help you test your knowledge and problem-solving skills.
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NCERT solutions of Quadrilaterals Class 9 Maths - all questions of all exercises are solved. Q. No. 1, 2, 3, and so on. Complete explanation and detailed solution makes them the best videos for Class 9 Maths Quadrilaterals NCERT solutions.