Question

M, N and T are the midpoints of the sides of $\Delta \text{XYZ}$.

If $\text{XZ}=10$, then $\text{MN}=\underset{¯}{?}$

Short answer

The length of MN is 5 units.

Step-by-step solution

Step 1. Consider the diagram.

$\Delta XYZ$is:

Here, M, N and T are midpoints of $\Delta XYZ$. and $XZ=10$ .

Step 2. Show the calculation.

Consider in triangles TXM and ZXY.

$\angle TXM=\angle ZXY$ (Common).

$\angle TXM=\angle ZXY$ (Midpoints).

Therefore, $\angle TXM=\angle ZXY$(SAS).

$\begin{array}{l}\angle XMT=\angle XYZ\\ \angle MTX=\angle YZX\end{array}$

Angle XMT and XYZ are the corresponding angles. Hence, line TM and ZY or ZN are parallel to each other.

Consider in triangles NMY and ZXY .

$\angle NYM=\angle ZYX$ (Common).

$\frac{NY}{ZY}=\frac{YM}{YX}=\frac{1}{2}$ (Midpoints).

Therefore, $\Delta NMY~\Delta ZXY$(SAS).

$\begin{array}{l}\angle YNM=\angle YZX\\ \angle YMN=\angle YXZ\end{array}$

Angle YNM and YZX are the corresponding angles. Hence, line NM and ZX or ZT are parallel to each other.

Thus, MNZT is parallelogram.

$ZT=MN$

The side MN is calculated as follows:

$\begin{array}{c}MN=ZT\\ =\frac{XZ}{2}\\ =\frac{10}{2}\\ =5\text{units}\end{array}$

Step 3. State the conclusion.

Therefore, the length of MN is 5 units.