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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?

Respuesta :

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

[tex]WZ^2=ZX\times ZY[/tex]

[tex](k+4)^2=(k+12)k[/tex]

[tex]k^2+8k+16=k^2+12k[/tex]

[tex]8k+16=12k[/tex]

Subtract both sides by 8k.

[tex]16=12k-8k[/tex]

[tex]16=4k[/tex]

Divide both sides by 4.

[tex]4=k[/tex]

Now,

[tex]XZ=12+k[/tex]

[tex]XZ=12+4[/tex]

[tex]XZ=16[/tex]

Therefore, the measure of XZ is 16 units.

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