Problem: Triangle ABC is inscribed in a semicircle of radius r. 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:
Answer Choices:
A. s2≦8r2
B. s2=8r2
C. s2≧8r2
D. s2≦4r2
E. s2=4r2
Solution:
Let AC=h and BC=l. Then h2+l2=4r2 Since s=h+l,s2=h2+l2+2hl=4r2+2hl=4r2+4 times the area of △ABC. The maximum area of △ABC occurs when h=l=r2. ∴s2≦4r2+4(r2)=8r2.