Midpoint HELP!!!!!!!!!! (proofs)

Question

In ΔABC, the points M and N lie 5/8 of the way down the sides on which they lie. 

 

Prove: 

i.  Line Segment MN is parallel to the base Line Segment BC

ii. Length MN is 5/8 the length of BC

Answer 1

The sides are in proportion which proves similarity...

Specifically triangle AMN is similar to triangle ABC

and AMN = (5/8) ABC with respect to the side measures..

Proof: Let x=AB and y = AC

then AM = (5/8)x and AN = (5/8)y

AM/AB = (5/8)X over x = 5/8

and

AN/AC = (5/8)y over y = 5/8

This proves similarity via SAS since angle

BAC and angle MAN are the same

Corresponsing angles of similar triangles are equal, so

angles AMN and ABC are equal, as are angles ANM and ACB.

This proves that MN is parallel to BC.

Finally, MN and BC must respect the same similarity

ratio so MN = (5/8) BC