Question

A circle is shown. Secants R S and R T intersect at point R outside of the circle. Secant R S intersects the circle at point U. Secant R T intersects the circle at point V. The length of R U is 6, the length of U S is 10, and the length of R V is 8.
If secant segments SR and TR intersect at point R, find the length of VT.

Start by relating the secants and segments theorem to this diagram:

(RS)(
) = (
)(RV)

Substitute values from the diagram into the equation:

(16)(
) = (
)(8)

Solve for VT:

VT =

Answer 1

Use the secant-secant theorem: (RS)(RU) = (RT)(RV).

Substitute RS = 16, RU = 6, RV = 8, and RT = 8 + VT:

(16)(6) = (8 + VT)(8)

96 = 64 + 8VT
32 = 8VT
VT = 4