Question

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

State how many parallelograms are in the diagram.

Short answer

There are 3 parallelograms in the diagram.

Step-by-step solution

Step 1. Consider the diagram.

$\Delta XYZ$is:

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

Step 2. Show the calculation.

Consider in triangles TXM and ZXY .

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

$\frac{TX}{ZX}=\frac{XM}{XY}=\frac{1}{2}$ (Midpoints).

Therefore, $\Delta TXM~\Delta ZXY$(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.

Similarly, TNXM is also a parallelogram.

Similarly, TNMY is also a parallelogram

Step 3. State the conclusion.

Therefore, the total number of parallelograms is 3.