Chapter 10.5, Problem 38HP

To determine

To explain: The segments $\overline{XY},\overline{XZ},\overline{XW}$ are congruent or not if the circle has different radii.

Answer to Problem 38HP

Yes, the segments $\overline{XY},\overline{XZ},\overline{XW}$ are congruent if the circle has different radii.

Explanation of Solution

Given information:

  

From figure, get that: two circles have the different radius. The lengths of tangents from an external point to a circle are both equal.

The line $\overline{XZ}$ is tangent to both the circles from the point $X$ . The line $\overline{XY}$ is tangent to the circle $\odot A$ from the same point $X$ .

Therefore,

  $\overline{XY}\cong \overline{XZ}$............. (1)

Similarly, the line $\overline{XW}$ is tangent to the circle $\odot B$ from the same point $X$ .

Therefore,

  $\overline{XW}\cong \overline{XZ}$............. (2)

Combining two equations (1) and (2):

  $\overline{XY}\cong \overline{XZ}\cong \overline{XZ}$

Therefore, the lines are congruent.