Problem:
In triangle , and . Let denote the midpoint of and let denote the intersection of with the bisector of angle . Which of the following is closest to the area of the triangle ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
By Heron's Formula the area of triangle is , which is 84 , so the altitude from vertex is . The midpoint divides into two segments of length 7 , and the bisector of angle divides into segments of length and (since the angle bisector divides the
opposite side into lengths proportional to the remaining two sides). Thus the triangle has base and altitude 12 , so its area is .
The problems on this page are the property of the MAA's American Mathematics Competitions