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.

(1 point)
Responses

x=169144

x equals Start Fraction 169 over 144 End Fraction

x=12

x equals 12

x=16912

x equals Start Fraction 169 over 12 End Fraction

x=1698

Answer 1

Let BC = x. Using the altitude-to-hypotenuse relations from the Pythagorean-proof:

AC^2 = BC·DC → 13^2 = x·DC so DC = 169/x.
AD^2 = BD·DC → 5^2 = BD·DC.

But BD = x − DC = x − 169/x. So
25 = (x − 169/x)(169/x) = 169(x^2 − 169)/x^2.

Multiply through: 25x^2 = 169(x^2 − 169) = 169x^2 − 169^2.
So 144x^2 = 169^2, hence x^2 = 169^2/144 and x = 169/12.

Answer: x = 169/12.