Question

In triangle ABC A B C , E E is the midpoint of AB⎯⎯⎯⎯⎯ A B ¯ , and G G is the midpoint of BC⎯⎯⎯⎯⎯ B C ¯ . A triangle ABC and line segment EG is shown. Select all the true statements. The length of EG E G is half the length of AB A B . The length of EG E G is half the length of BC B C . The length of EG E G is half the length of AC A C . Line segment EG E G is parallel to line segment AC A C . Line segment EG E G is perpendicular to line segment AB A B .

Answer 1

The length of EG is half the length of AB, but it is not necessarily half the length of BC or AC. EG is not inherently parallel to AC or perpendicular to AB.

In triangle ABC, where E is the midpoint of AB and G is the midpoint of BC, several statements can be assessed.

1. The length of EG is half the length of AB:

True. By the midpoint property, E is the midpoint of AB, implying that AE = EB. Similarly, G is the midpoint of BC, indicating that BG = GC. Therefore, EG = AE + BG = EB + GC = AB. Thus, the length of EG is indeed half the length of AB.

2. The length of EG is half the length of BC:

False. While EG is equal to the sum of AE and BG, which are both equal to half of AB, it does not necessarily imply that EG is half the length of BC. The relationship between EG and BC is not directly established by the midpoint property.

3. The length of EG is half the length of AC:

False. Similar to the second statement, the midpoint property does not directly relate EG to AC. The length of EG is not guaranteed to be half the length of AC.

4. Line segment EG is parallel to line segment AC:

False. Without additional information or specific properties mentioned, the parallelism between EG and AC cannot be determined solely based on the midpoint property.

5. Line segment EG is perpendicular to line segment AB:

False. The midpoint property does not imply any perpendicular relationship between EG and AB. Perpendicularity would require additional geometric information or constraints.