Question

$\overline{RE}$is an altitude of $\Delta RST$.Find$MN,NE,RT$ .

Short answer

$MN=14,NE=15,RT=26$

Step-by-step solution

Step 1- Draw the diagram.

Consider the diagram.

Step 2- Step description.

Consider the triangle$△RST$.

Here, points Mand N are the mid points of$\overline{RT}$and $\overline{RS}$.

So, the side $\overline{MN}$is half as long as the third side$\overline{ST}$.

$\begin{array}{c}MN=\frac{1}{2}ST\\ =\frac{1}{2}(SE+ET)\\ =\frac{1}{2}(18+10)\end{array}$

$\begin{array}{c}=\frac{1}{2}\left(28\right)\\ =14\end{array}$

Step 3- Step description.

Consider the triangle $△RST$.

It is given that $\overline{RE}$ is an altitude of $△RST$.

So, it can be said that the triangles $△SER$and $△TER$ are right triangles.

Now,$\overline{RS}$ is the hypotenuse of the right triangle $△SER$ and N is its midpoint.

So, the midpoint N of the hypotenuse $\overline{RS}$ of a right triangle $△SER$ is equidistant from the three vertices.

Thus,$NE=NR=15$ .

Step 4- Step description.

Similarly,$\overline{RT}$ is the hypotenuse of the right triangle$△TER$ and M is its midpoint.

So, the midpoint N of the hypotenuse $\overline{RT}$ of a right triangle $△TER$is equidistant from the three vertices.

Thus, $MT=ME=13$.

Also, M is the midpoint of $\overline{RT}$. So, it implies that,$RM=MT=13$ .

Therefore,

$\begin{array}{c}RT=RM+MT\\ =13+13\\ =26\end{array}$

Therefore,$MN=14,RT=26,NE=15$ .