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 $\text{SA}$.

Short answer

$\text{SA}=\mathbf{15}$

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.

Consider the triangle $\Delta SAT$.

It is given that $\overline{RA}$is an altitude of $\Delta SAT$.

So, it can be said that the triangle $\Delta ARS$is a right triangle.

Now, $\overline{SA}$ is the hypotenuse of the right triangle $\Delta ARS$ and P is its midpoint.

So, by theorem 5-15, the midpoint P of the hypotenuse $\overline{SA}$of a right triangle $\Delta ARS$is equidistant from the three vertices.

Thus,

$SP=PR=7.5$

Also, P is the midpoint of $\overline{SA}$. So, it implies that,

$SP=PA$

Therefore,

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

Step 3. State the conclusion.

Therefore, $SA=15$