jmc

Question

ABC is inscribed in a semicircle of radius r so that its base AB coincides with diameter AB. Point C does not coincide with either A or B. Let s=AC+BC. Then, for all permissible positions of C: (A) s^28r^2 (B) s^2=8r^2 (C) s^2 8r^2 (D) s^24r^2 (E) s^2=4r^2

Accepted Answer

FIGURESince s=AC+BC, s^2 = AC^2 + 2 AC BC + BC^2. Since ABC is inscribed and AB is the diameter, ABC is a right triangle, and by the Pythagorean Theorem, AC^2 + BC^2 = AC^2 = (2r)^2. Thus, s^2 = 4r^2 + 2 AC BC. The area of ABC is AC BC2, so 2 [ABC] = AC BC. That means s^2 = 4r^2 + 4 [ABC]. The area of ABC can also be calculated by using base AB and the altitude from C. The maximum possible value of the altitude is r, so the maximum area of ABC is r^2. Therefore, s^2 8r^2