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Question

State true or false:


In quadrilateral ABCD, the diagonals AC and BD intersect each each other at right angle, then AB2+CD2=AC2+DA2

A
True
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B
False
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Solution

The correct option is B False
Given: Quadrilateral ABCD, diagonals intersect at right angles at O
In OAB,
OA2+OB2=AB2 (I) (Pythagoras theorem)
In OBC,
OB2+OC2=BC2 (II) (Pythagoras theorem)
In OCD,
OC2+OD2=CD2 (III) (Pythagoras theorem)
In OAD,
OA2+OD2=DA2 (IV) (Pythagoras theorem)
Add I and III,
OA2+OB2+OC2+OD2=AB2+CD2 (V)
Add II and IV,
OA2+OB2+OC2+OD2=BC2+DA2 (VI)
Equating V and VI,
AB2+CD2=BC2+DA2

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