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# Geometry Problem

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Right triangle ABC has one leg of length 6 cm, one leg of length 8 cm, and a right angle at A. A square has one side on the hypotenuse of triangle ABC and a vertex on each of the two legs of triangle ABC. What is the length of one side of the square in cm? Express your answer as a common fraction.

Feb 13, 2018

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Here's my solution

AB  = 6, BC   = 10 and  AC  = 8

Call the point where the upper left vetex of the triangle meets the triangle  = D

Call the point where the upper right vertex of the triangle meets the triangle = F

Call the point where the bottom left vertex of the square meets the triangle = E

And we have three congruent triangles   ABC  ~ ADF  ~ EBD

Let  the side of the square  = x = DF    and let AD  = y

And we have that

DF / AD  = BC / AB

x / y   =   10/6     ⇒  y = (3/5)x

And

BD / DE   = BC / AC

[ 6 - y ] / x  =  10 / 8        subbing for y, we have that

[6  -  (3/5)x ] / x  =  10 / 8  =  5/4

Cross-multiply

4 [ 6  - (3/5)x ]  = 5x

24  -  (12/5)x  =  5x

24  =  5x +  (12/5)x

24  =  [ 25  + 12]x / 5

120  = 37 x

x  ⇒  120 / 37    =   side of the square

Feb 14, 2018
edited by CPhill  Feb 14, 2018
edited by CPhill  Feb 14, 2018