Chapter 5.3, Problem 9WE

To determine

To name: points that must be the midpoints of the sides of the large triangle

Answer to Problem 9WE

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

Explanation of Solution

  

In $\Delta ABC$ ,

  $CF=FB$ (Given)

So,

F is a 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 $CF=FB$ , $\frac{CF}{FB}=1$

So,

  $1=\frac{CD}{DA}\Rightarrow CD=DA$

So,

D is a midpoint of AC.

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