Chapter 5.4, Problem 27WE

To determine

To Find : The Perimeter of $\Delta PQR$ use condition.

Answer to Problem 27WE

Perimeter is 60.

Explanation of Solution

Given information:

NA is an altitude of ΔSAT.

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,

  $AQ=QT=10$

From segment addition postulate,

  $\begin{array}{l}AQ+QT=AT\hfill \\ AT=10+10\hfill \\ =20\hfill \end{array}$

In $\Delta ARS,P$ is the midpoint of $\overline{SA}$ .

  $\begin{array}{l}SP=PA=PR\hfill \\ SP=PA=7.5\hfill \end{array}$

From segment addition postulate.

  $\begin{array}{l}\begin{array}{l}SP+PA=SA\hfill \\ SA=7.5+7.5\hfill \end{array}\\ =15\end{array}$

From segment addition postulate

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

In $\Delta AST$ , Perimeter is

  $\begin{array}{l}\begin{array}{l}=AS+ST+AT\hfill \\ =15+25+20\hfill \end{array}\\ =60\end{array}$

Hence perimeter is 60.