Question

A circle is shown. Secant X Z and tangent W Z intersect at point Z outside of the circle. Secant X Z goes through the center of the circle and intersects the circle at point Y. The length of W Z is k + 4, the length of Z Y is k, and the length of X Y is 12.

What is the length of line segment XZ?

Answer 1

Given:

Consider the below figure attached with this ques.

To find:

The length of the line segment XZ.

Solution:

According to the tangent-secant theorem, the square of tangent is equal to the product of secant and external segment of secant.

Using tangent-secant theorem, we get

`WZ^2=ZX* ZY`

`(k+4)^2=(k+12)k`

`k^2+8k+16=k^2+12k`

`8k+16=12k`

Subtract both sides by 8k.

`16=12k-8k`

`16=4k`

Divide both sides by 4.

`4=k`

Now,

`XZ=12+k`

`XZ=12+4`

`XZ=16`

Therefore, the measure of XZ is 16 units.