Question

A circle is shown. Secants E C and A C intersect at point C outside of the circle. Secant E C intersects the circle at point D. Secant A C intersects the circle at point B. The length of E D is 14, the length of D C is x + 1, the length of A B is 21, and the length of B C is x.
What is the value of x?

x = 2
x = 3
x = 4
x = 6

Answer 1

Use the secant-secant power theorem: (external segment)*(whole secant) are equal for both secants from C.

For secant C–D–E: external = DC = x+1, whole = DC+DE = (x+1)+14 = x+15.
For secant C–B–A: external = CB = x, whole = CB+BA = x+21.

Set equal:
(x+1)(x+15) = x(x+21)
x^2 + 16x + 15 = x^2 + 21x
15 = 5x
x = 3

Answer: x = 3.