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A square is inscribed in a right triangle, as shown below. The legs of the triangle are 1 and 3. Find the area of the square.

 

 Jan 11, 2024
 #1
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Triangles CDE  and EFB  are similar

 

So

 

CD / DE  =  EF / FB

 

(1 - x)  / x   =  x / (3 - x)             cross-multiply

 

(3 - x) ( 1 -x)   = x^2

 

x^2  - 4x + 3   =   x^2

 

-4x + 3  = 0

 

3  = 4x

 

x  = 3/4

 

Area of  square  = x^2  =   (3/4)^2  =  9 / 16

 

Notice something kinda' neat   

 

Side of square  =  x =  (product of   legs)  / (sum of legs) =  (3 * 1) / ( 3 + 1)  = 3 / 4

 

cool cool cool

 Jan 11, 2024
edited by CPhill  Jan 11, 2024

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