Chapter 10.6, Problem 26PSC

To determine

Give the explanation for the given proof.

Explanation of Solution

Given:

The given statement is that two circles are internally tangent and the center of the larger circle is on the smaller circle.

Calculation:

Draw a diagram using given data.

  

The angle inscribed in a semicircle is a right angle.

  $m\angle PRQ=m\angle OSQ={90}^{°}$

The angle $\angle PRQ$ and $\angle OSQ$ are the corresponding angle.

So,

  $PR\parallel OS$

The segment $\overline{OS}$ divides the two sides into proportional segments.

  $\frac{QS}{SR}=\frac{OQ}{OP}=1$ , because $OQ=OP$ (radius)

  $\Rightarrow QS=SR$ .

Hence the chord that has one endpoint at the point of tangency is bisected by the smaller circle