Chapter 15, Problem 43CR

To determine

To prove: that the triangles formed by two tangents, if two tangents are drawn to a circle from an exterior point and any tangent to the minor arc included by them has a perimeter equal to the measures of two original tangents segments.

Answer to Problem 43CR

The perimeter equal to the measures of two original tangents segments.

Explanation of Solution

Given:

The two tangents are drawn to a circle from an exterior point as shown in the given figure.

  

Calculation:

According to the question, the two tangents are drawn to a circle from an exterior point and any tangent to the minor arc included by them has a perimeter equal to the measures of two original tangents segments. Then,

  $\begin{array}{l}CA\cong CD\{\text{by two-tangent theorem}}\\ BE\cong BD\{\text{by two-tangent theorem}}\end{array}$

Therefore, $PC+CD=PA\text{ and }PE+ED=PB$ .

Now,

  $\begin{array}{l}PC+\left(CD+ED\right)+PE=PA+PB\\ PC+CE+PE=PA+PB\end{array}$

Hence, the perimeter equal to the measures of two original tangents segments.