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Right triangle ABC has side lenghts AB = 3, BC = 5. Square XYZW is inscribed in triangle ABC with X and Y on line AB, and Z on line BC. What is the side length of the square.

 

 Jul 29, 2022
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angle ABC  = 90°

 

Note that AC = sqrt (3^2 + 5^2)  = sqrt (34)

 

Area of ABC =  (1/2) (AB) (BC)  = (1/2) (3) (5)  = 7.5

 

Height of ABC  can be found as

 

2* area of ABC / AC   =  2 * 7.5  /  sqrt (34)  =     15 / sqrt (34)

 

And triangles  ABC and  WBZ are similar

 

Call the side of the square  S

 

Base of ABC / Height of ABC =  Base of WBZ / Height of WBZ

 

sqrt (34) / ( 15 /sqrt (34) ) =  S  / (15/sqrt (34)  - S)

 

34 / 15  = S / ( 15/sqrt (34) - S)

 

(34/15) ( 15 /sqrt (34) - S)  =  S

 

sqrt (34)  - ( 34/15)S = S

 

sqrt (34) = S ( 1 + 34/15)

 

S =  sqrt (34)  / ( 1 + 34 /15)  =  15 *  sqrt (34)  / ( 49)  =  (15/49)sqrt (34) ≈ 1.785

 

 

cool cool cool

 Jul 29, 2022

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