Question

3 connecting lines are shown. Line D F is horizontal. Line D E is about half the length of line D F. Line F E is about one-third of the length of line D F.
Which inequality explains why these three segments cannot be used to construct a triangle?

EF + FD > DE
ED + EF < DF
ED + EF > DF
EF + FD < DE

Answer 1

To determine why the three segments cannot be used to construct a triangle, we must apply the triangle inequality theorem. The triangle inequality states that, for any three sides of a triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

Given the segments:
- \( DE \) (half the length of \( DF \))
- \( EF \) (one-third the length of \( DF \))
- \( DF \) (horizontal line)

If we represent \( DF \) as \( x \), then:
- \( DE = \frac{1}{2}x \)
- \( EF = \frac{1}{3}x \)

Now we check the conditions for forming a triangle:

1. **Check \( DE + EF > DF \)**:
\[
DE + EF = \frac{1}{2}x + \frac{1}{3}x = \frac{3}{6}x + \frac{2}{6}x = \frac{5}{6}x
\]
Here, \( DE + EF = \frac{5}{6}x < DF = x \).

2. **Check \( DE + DF > EF \)**:
\[
DE + DF = \frac{1}{2}x + x = \frac{3}{2}x > EF = \frac{1}{3}x
\]
This condition holds true.

3. **Check \( EF + DF > DE \)**:
\[
EF + DF = \frac{1}{3}x + x = \frac{4}{3}x > DE = \frac{1}{2}x
\]
This condition holds true as well.

The inequality that fails is:
\[
DE + EF < DF
\]

Thus, the correct choice that explains why these three segments cannot be used to construct a triangle is:

**ED + EF < DF**