Chapter 5.4, Problem 25WE

To determine

To find: $SA$

Answer to Problem 25WE

  $SA=15$

Explanation of Solution

Given:

The diagram:

$\overline{RA}$ is an altitude of $△SAT$ .

P and Q are midpoints of $\overline{SA}$ and $\overline{TA}$ .

Consider the figure shown below:

Consider the triangle $△SAT$ .

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

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

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

So, by theorem 5-15, the midpoint P of the hypotenuse $\overline{SA}$ of a right triangle $△ARS$ 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,

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