Question

HELP QUICK. I WILL THUMBS UP

In triangle ABC shown below, side AB is 8 and side AC is 4:

Triangle ABC with segment joining point D on segment AB and point E on segment AC.

Which statement is needed to prove that segment DE is half the length of segment BC?

Segment AD is 4, and segment AE is 8.
Segment AD is 2, and segment AE is 4.
Segment AD is 4, and segment AE is 2.
Segment AD is 2, and segment AE is 2.

Answer 1

Segment AD is 4, and segment AE is 2.

Explanation:

The given triangle is attached.

In this problem we have two proportional triangles. We know that because it can be applied the theorem of Thales, which is about similarities, it describes the ratio between proportional segment, and this happens when we have a pairs of parralles crossed by transversal, as we have here.

So, if AB = 8 and AC = 4, which belong to the bigger triangle. That means the smaller must be in the same ratio, that is, one side must be half the other one. SO, from this proportion we know that if AD = 4, then AE = 2, and this is the statement we would need to prove that DE is half of the length of BC.

Actually, from the given graph and the given values, we already know that the ratio of this proportional figures is 1/2.

Therefore, the right answer is "Segment AD is 4, and segment AE is 2", because it represents the same proportion that the given one, and gives the proportion we want to find.

Answer 2

A diagram would be nice. The quick answer is C.

The first answer makes AE 8 which is twice as long as AC. I don't think the intent of the question is to use an external point.

The second one is also incorrect. This time E is in a place such that it makes AC and AE both equal and that too isn't possible.

The third one is the answer because D and E are both 1/2 the distance between the end point of the line containing them.

D is wrong because E is the midpoint of AC but D is not the midpoint of AB.