Let $\overline{XY}$ be a tangent to a circle, and let $\overline{XBA}$ be a secant of the circle, as shown below. If $AX = 10$ and $XY = 8$, then what is $AB$?

kittykat Mar 20, 2024

#1**+1 **

We can use the Tangent-Secant Theorem to solve for AB.

The Tangent-Secant Theorem states that the square of the tangent segment (XY in this case) is equal to the product of the whole secant segment (XB) and the external secant segment (XA).

In this case, we are given that XY=8 and XA=10. Let XB=b.

Therefore, applying the Tangent-Secant Theorem:

XY2=XB⋅XA

82=b⋅10

64=10b

b=1064

b=6.4

Therefore, the length of segment AB is $ \boxed{6.4}$.

Boseo Mar 20, 2024