Question

The following sides of triangle upper A upper C upper B are labeled: upper A upper C is 13, and upper B upper C is lower x. Upper A upper D is labeled 5. Right angle symbols are drawn at angle upper B upper A upper C and angle upper A upper D upper C.

Use the steps from the proof of the Pythagorean Theorem to find x.

Answer 1

Place D as the foot of the altitude from A to the hypotenuse BC. From the similarity relations used in the Pythagorean proof we have

1) AC^2 = BC · DC, so DC = AC^2 / BC = 13^2 / x = 169/x.

2) AD^2 = DC · BD = DC · (BC − DC) = DC(x − DC). With AD = 5 this gives
25 = (169/x)·(x − 169/x).

Multiply through by x^2:
25x^2 = 169(x^2 − 169) = 169x^2 − 169^2.

Rearrange:
144x^2 = 169^2 = 28561 => x^2 = 28561/144 => x = 169/12.

So x = 169/12.