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.
+130081 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
