Chapter 9.7, Problem 26WE

To determine

To prove: that $T$ is the midpoint of $\overline{PQ}$

Answer to Problem 26WE

  $T$ is the midpoint of $\overline{PQ}$ is proved

Explanation of Solution

Given information:

The common chord extended to meet $\overline{PQ}$ at point $T$

A line is tangent to two intersecting circles at $P$ and $Q$

Calculation:

Let lbe the common tangent of the two circles that intersect.

The common chords $\overline{AB}$ meet $\overline{PQ}$ at point $T$

Consider the following theorem

When a segment and a tangent segment are drawn from an outer point to a circle, the product of the secant segment and its outer segment is equal to the square tangent segment.

Using the theorem and solve,

Since, $\overline{TA}$ is the secant and $\overline{TP}$ is the tangent to first circle as shown in the figure

So,

  $TA\cdot TB=T{P}^2$

Similarly, $\overline{TA}$ is the secant and $\overline{TQ}$ is the tangent to first circle as shown in the figure

So,

  $TA\cdot TB=T{P}^2$

Similarly, $\overline{TA}$ is the secant and $\overline{TQ}$ is the tangent to first circle as shown in the figure

So,

  $TA\cdot TB=T{Q}^2$

Now, make the two equations equal:

  $T{P}^2=T{Q}^2$

Take square root on both sides.

  $\sqrt{T{P}^2}=\sqrt{T{Q}^2}$

  $TP=TQ$

Since $TP=TQ,$ then $T$ is the midpoint of $\overline{PQ}$

Hence, it is proved that $T$ is the midpoint of $\overline{PQ}$