Question

In triangle ABC shown below, side AB is 8 and side AC is 6: 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 3, and segment AE is 4. Segment AD is 3, and segment AE is 6. Segment AD is 4, and segment AE is 6. Segment AD is 4, and segment AE is 3.

Answer 1

For the segment DE to be the midsegment of the triangle ABC, the point D must cut AB in two equal halves, and point E must cut AC in two equal halves as well, that way, DE touches both ends right in the middle

and that can only occur if AD is 4 (half of 8) and AE is 3 ( half of 6)

Answer 2

Well, the only way I figured to use was using the method for 'similar triangles'
So, what I did was, since the whole of AB is similar to AD, and the whole of AC is similar to AE...

AB/AD =AC/AE 

Then, since We already know AB and AC, i'll choose one as the 'unknown' to check my answer, so I chose AC i this case which has a value of 6...(You'll know what i mean in a bit) and keep substituting the values of AD and AE to check if my 'unknown' is indeed 6//
So for the first statement: 
8/3 = AC/4 
 ( solve for AC ) == 8 x 4 / 3 
AC =10.6...which we know is not the value of AC, as it is 6

Second statement
8/3 =AC /6 
AC =16 xxx (wrong)

Third statement

8/4 = AC/6
AC = 12 xxx

Fourth Statement
8/4 =AC/3
AC = 6, which is what we have!

So I THINK the last statement is right
P.s## I'm still a little skeptical about my method, but nonetheless hope this helps