Question

YZ
is tangent to ⨀X. What is XY?
X
Y
Z
5 mm
3 mm
XY=
mm

Answer 1

To find the length of segment \( XY \) when line \( YZ \) is tangent to circle \( \odot X \) at point \( Y \), we can use the Power of a Point Theorem. According to this theorem, the length of the tangent segment from a point outside the circle to the point of tangency is equal to the square root of the power of that point relative to the circle.

Given the following:
- \( YZ \) is tangent to circle \( \odot X \) at point \( Y \).
- The distance from point \( X \) (the center of the circle) to point \( Y \) is given as \( XY = r \), where \( r \) is the radius of the circle.
- The length of the tangent segment \( YZ = 5 \) mm.
- The length of the line segment from the center of the circle to the point where the tangent meets the circle at \( Y \) is \( 3 \) mm.

Using the information:
1. The tangent length \( YZ \) is \( 5 \) mm.
2. The radius \( XY \) is \( 3 \) mm.

According to the Power of a Point theorem:
\[
YZ^2 = XY^2
\]

Rearranging gives:
\[
XY = \sqrt{YZ^2} = \sqrt{5^2} = 5 \text{ mm}
\]

Thus, the length \( XY \) is:
\[
XY = 3 + 5 = 8 \text{ mm}
\]

Therefore, the solution is:
\[
XY = 8 \text{ mm}
\]