Question

Name all the points shown that must be midpoints of the sides of the large triangle.

Short answer

D and F must be the midpoints of the side AC and CB of the large triangle.

Step-by-step solution

Step 1. Consider the diagram.

Consider the large triangle ABC.

Step 2. State the explanation.

In $\Delta ABC$,

$CF=FB$ (Given)

So,

F is the midpoint of BC.

(Midpoint is a point on a line segment that divides it into two equal parts.)

Also,

$DF\parallel AB$ (Given)

According to Basic Proportionality Theorem,

If a line is drawn parallel to one side of a triangle such that it intersects the other sides in different points, then the other two sides are divided in the same ratio.

So,

$\frac{CF}{FB}=\frac{CD}{DA}$

As $CF=FB$, then,

$\frac{CF}{FB}=1$

So,

$\begin{array}{c}1=\frac{CD}{DA}\\ CD=DA\end{array}$

Thus, D is a midpoint AC.

Step 3. State the conclusion.

Therefore, D and F must be the midpoints of the side AC and CB of the large triangle.