Question

If the dialogues of a quadrilteral interesect each other at right angles then prove that
the sum of square of opposite sides are equal.
4 marks question​

Answer 1

Given :-

• A quadrilateral is given

• The diagonals of a quadrilateral intersect each other at right angles that is 90°

To Prove :-

• Sum of square opposite sides are equal

Proof :-

Let the quadrilateral be ABCD

Now , In ΔAOB ,

Angle AOB = 90°

[ Diagonals intersect each other at 90° ]

Therefore,

By using Pythagoras theorem, we get

AB² = AO² + OB² ...eq( 1 )

Similarly, In ΔCOD

By using Pythagoras theorem, we get

CD² = CO² + DO² ....eq( 2)

Now, Adding eq( 1) and ( 2)

AB²+ CD² = AO² + OB² + CO² + DO² ...eq( 3)

Now, In AOD, Angle AOD = 90°

By using Pythagoras theorem,

AD² = AO² + DO² ...eq( 4)

Similarly, In ΔCOB

By using Pythagoras theorem, we get

BC² = BO² + CO² ...eq( 5)

Adding eq( 4) and ( 5)

AD² + BC² = AO² + DO² + BO² + CO² ...eq( 6)

Now, From eq( 3) and eq( 6) we get,

AB² + CD² = AD² + BC²

Hence, Proved $.$

[ Note :- Refer attachement for better understanding ]