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Quadrilateral has right angles at B and D, and AC = sqrt(65). If ABCD has distinct integer side lengths, then what is the area of ABCD?

 Nov 16, 2020
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We  need to find two Pythagorean Triples here such that

 

a^2 + b^2   = 65

 

There are two  pairs that work    (8,1)  and ( 7, 4)

 

So.....we can let  AB = 8 ,BC = 1, CD = 7  and DA  = 4

 

We have  two right triangles  sharing the same  hypotenuse  ( AC)

 

The area of the quadrilateral is   (1/2) [ (8) (1)  + (7) (4) ]  =  (1/2) [ 8 + 28 ]  = (1/2) (36)  = 18

 

 

cool cool cool  

 Nov 16, 2020

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