In triangle ABC, points M, N, P mark the midpoints of sides AB, BC, CA, respectively.

In triangle ABC, points M, N, P mark the midpoints of sides AB, BC, CA, respectively. Find the perimeter of rectangle MNP if AB = 8 BC = 10 CA = 12

Consider a triangle ABC, its sides AB = 8, BC = 10, CA = 12, construct points: M is the middle of AB, N is the middle of BC, P is the middle of AC. We got a triangle МNР, let’s find its perimeter.
Let’s compare triangles ABC and PNC, <PCN = <ACB, AC / PC = BC / NC = 2, which means that the triangles are similar in two sides and the angle between them, which means:
AC / PC = BC / NC = 2 = AB / PN, hence:
PN = AB / 2 = 8/2 = 4.
Let’s compare triangles ABC and PMA, <PAM = <CAB, AC / PA = BA / AM = 2, which means that the triangles are similar in two sides and the angle between them, which means:
AC / PA = BA / AM = 2 = BC / MP, hence:
MP = BC / 2 = 10/2 = 5.
Let’s compare triangles ABC and NMB, <NBM = <CBA, AB / BM = BC / BN = 2, which means the triangles are similar in two sides and the angle between them, which means:
AB / BM = BC / BN = 2 = CA / MN, hence:
MN = CA / 2 = 10/2 = 6.
The perimeter of the MNP triangle is:
p = PN + MP + MN = 4 + 5 + 6 = 15 cm.
Answer: the perimeter of MNP is 15 cm.