Question

in a quadrilateral abcd angle b=90 if ad2=ab​2+bc2+cd2.prove that angle acd =90

Answer 1

ABCD quadrilateral.    angle ABC = 90 deg.
               hence,  AB² + BC² = AC²     --- (1)    using Pythagoras theorem.

AD² = ( AB² + BC² ) + CD²  given,
     =  AC² + CD²          ---- (2)

In the triangle ACD,  AD has to be a hypotenuse and angle ACD must be 90 deg. as per Pythagoras theorem.

Answer 2

Given: ABCD is a quadrilateral, ∠B = 90° and AD2 = AB2 + BC2 + CD2

To prove: ∠ACD = 90°

Proof: In right ∆ABC,

AC2 = AB2 + BC2 … (1)

Given, AD2 = AB2 + BC2 + CD2

⇒ AD2 = AC2 + CD2 (Using (1)

In ∆ACD,

AD2 = AC2 + CD2

∴ ∠ACD = 90° (Converse of Pythagoras theorem)

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