Question

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

State the theorem suggested by Exercise 6.

Short answer

The theorem stated by Exercise 6 is the Midpoint Theorem.

Step-by-step solution

Step 1. Consider the diagram.

$\Delta XYZ$is:

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

Step 2. State the explanation.

Midpoint Theorem:

If the line joining midpoints of any triangle is parallel to the third side of that triangle, then this line is equal to the half of the third side of the triangle.

Now in $\Delta XYZ$,

Since T and N are the midpoints of ZX and ZY respectively.

So, according to Midpoint Theorem,

TN is parallel to XY and $TN=\frac{1}{2}XY$.

Since N and M are the midpoints of ZY and YX respectively.

So, according to Midpoint Theorem,

NM is parallel to ZX and $NM=\frac{1}{2}ZX$.

Since M and T are the midpoints of YX and XZ respectively.

So, according to Midpoint Theorem,

MT is parallel to YZ and $MT=\frac{1}{2}YZ$ .

Step 3. State the conclusion.

Therefore, the theorem stated by Exercise 6 is the Midpoint Theorem.