Question

$\overline{\text{RA}}$is an altitude of $\Delta \text{SAT}$. Pand Q are midpoints of $\overline{\text{SA}}$and $\overline{\text{TA}}$. $\text{SR}=9$, $\text{RT}=16$, $\text{QT}=10$, and $\text{PR}=7.5$.

Find the perimeter of $\Delta \text{PQR}$.

Short answer

The perimeter of $\Delta \text{PQR}=\mathbf{30}$.

Step-by-step solution

Step 1. Consider the diagram.

Here, $\overline{RA}$is an altitude of $\Delta SAT$. P and Q are midpoints of $\overline{SA}$and $\overline{TA}$. $SR=9$, $RT=16$, $QT=10$, and $PR=7.5$.

Step 2. Show the calculation.

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,

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

Hence,

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

In $\Delta PQR$,

Perimeter is:

$\begin{array}{c}\text{Perimeter}=PQ+QR+PR\\ =12.5+10+7.5\\ =30\end{array}$

Step 3. State the conclusion.

Therefore, perimeter is 30.