O 24 In a quadrilateral ABCD, ZB = 90° = D. Prove that
2AC2 - BC2 = AB2 + AD2 + DC2.
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In a quadrilateral ABCD, ∠B=90, AD
2
=AB
2
+BC
2
+CD
2
, prove that ∠ACD=90
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ANSWER
As ∠ABC=90
∘
So applying Pythagoras theorem in △ABC
AB
2
+BC
2
=AC
2
(1)
Given: AD
2
=AB
2
+BC
2
+CD
2
(2)
Substituting (1) in (2)
AD
2
=AC
2
+CD
2
In △ACD , applying converse of Pythagoras theorem which states
that if the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.
Hence ∠ACD=90
∘
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