Chapter 5.4, Problem 26WE

To determine

To Find : The perimeter of ΔPQR.

Answer to Problem 26WE

The perimeter is 30.

Explanation of Solution

Given information:

  $\overline{RA}$ Is an altitude of $\Delta SAT$ .

P and Q are midpoints of $\overline{SA}$ and $\overline{TA}$ . $SR=9,RT=16,QT=10$ and $PR=7.5$ .

By definition of altitude

  $\overline{RA}\perp \overline{ST}$

The midpoint of the hypotenuse of the right triangle is equidistant from the three vertices.

In $\Delta ART$ , Q is the midpoint of $\overline{TA}$ .

So,

  $AQ=RQ=QT$

Hence,

  $RQ=10$

In $\Delta AST,P$ and $Q$ are midpoints of sides $\overline{AS}$ and $\overline{AT}$ and $\overline{ST}$ is the third side.

So,

  $PQ=\frac{1}{2}ST$

From segment addition postulate

  $\begin{array}{l}\begin{array}{l}ST=SR+RT\hfill \\ =9+16\hfill \end{array} \\ =25\end{array}$

Hence,

  $\begin{array}{l}PQ=\frac{25}{2}\hfill \\ =12.5\hfill \end{array}$

In $\Delta PQR$ ,

Perimeter is

  $\begin{array}{l}\begin{array}{l}=PQ+QR+PR\hfill \\ =12.5+10+7.5\hfill \end{array}\\ =30\end{array}$ .