Answer:
a) To show that AABC is similar to AMNC, we need to prove that their corresponding angles are congruent and their corresponding sides are proportional.
1. Angles: Since M is the midpoint of AC, we can see that angle AMB is congruent to angle CMB. Additionally, since N is the midpoint of BC, angle ANC is congruent to angle BNC.
2. Sides: Since M is the midpoint of AC, we know that AM = MC. Similarly, since N is the midpoint of BC, BN = NC.
Therefore, we have congruent angles (angle AMB ≅ angle CMB and angle ANC ≅ angle BNC) and proportional sides (AM/MC = BN/NC).
b) Let's denote the ratio of the sides of AMNC to AABC as k. From part a), we established that AM = MC and BN = NC.
Using the Scale Factor Property of Similar Triangles, we can say that:
AM/MC = BN/NC = k
Since AM = MC, k = 1. Similarly, BN = NC, so the ratio of the sides of AMNC to AABC is 1:1.
