Question

$\overline{\text{RA}}$is an altitude of $\Delta \text{SAT}$. P and 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 $\Delta \text{SAT}$.

Short answer

The perimeter of $\Delta \text{SAT}=\mathbf{60}$.

Step-by-step solution

Step 1. Consider the diagram.

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

$AQ=RQ=10$

From the segment addition postulate,

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

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

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

From the segment addition postulate,

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

From the segment addition postulate,

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

In $\Delta AST$,

Perimeter is:

$\begin{array}{c}\text{Perimeter}=AS+ST+AT\\ =15+25+20\\ =60\end{array}$

Step 3. State the conclusion.

Therefore, the perimeter is 60.