Question

UV and RV are secant segments that intersect at point V.

Circle C is shown. Secants U V and R V intersect at point V outside of the circle. Secant U V intersects the circle at point T, and secant R V intersects the circle at point S. The length of U V is 12, the length of T V is a, the length of R S is 5, and the length of S V is 4.

What is the length of TV?

1 unit
1Two-thirds units
2One-half units
3 units

Answer 1

Use the secant–secant power theorem: for two secants from an external point V, (external segment)·(whole secant) is equal for both.

For secant UV: external = TV = a, whole = VU = 12 → product = 12a.
For secant RV: external = SV = 4, inside piece SR = 5 so whole = VR = 4+5 = 9 → product = 4·9 = 36.

Set 12a = 36 → a = 3.

Answer: 3 units.