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 $\mathit{\text{RQ}}$.

Short answer

$RQ=10$

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 $\Delta ART$ is a right triangle.

Now, $\overline{TA}$is the hypotenuse of the right triangle $\Delta ART$ and Q is its midpoint.

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

Step 3. State the conclusion.

Therefore,$RQ=TQ=10$