Question

In the following figure is rhombus $\overline{RA}$is an altitude $\Delta SAT$. Pand Q are midpoint of $\overline{SA}$and $\overline{TA}$. $SR=9,RT=16,QT=10,PR=7.5$. Find $SA$.

Short answer

The value is $SA=15$.

Step-by-step solution

Step 1. Check the figure.

Consider the figure.

Step 2. Step description.

From the figure, $\overline{RA}$is an altitude of $\Delta SAT$.

Thus, $\Delta ARS$is a right triangle.

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

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

Step 3. Step description.

Thus, $SP=PR=7.5$.

Also P is the midpoint of $\Delta ARS$, then this implies that $SA=PA$.

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

Therefore, the value is $SA=15$.