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.

Guest Feb 13, 2018

#1**+2 **

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

CPhill Feb 14, 2018