Problem:
In △ABC,∠A=100∘,∠B=50∘,∠C=30∘,AH is an altitude, and BM is a median. Then ∠MHC=
Answer Choices:
A. 15∘
B. 22.5∘
C. 30∘
D. 40∘
E. 45∘
Solution:
Since ∠C=30∘ and AH⊥HC,∠CAH=60∘ and, in fact, △AHC is a 30-60-90 triangle. Hence, AH=21AC=AM. Thus, △AHM is equilateral, ∠AHM=60∘, and ∠MHC=90∘−60∘=30∘.
OR
Since M is the midpoint of hypotenuse AC of right triangle AHC,MH and MC are radii of the circle circumscribing △AHC. Therefore MH=MC and ∠MHC=∠C=30∘.
.jpg)