Problem:
A pyramid has a square base ABCD and vertex E. The area of square ABCD is 196 , and the areas of △ABE and △CDE are 105 and 91, respectively. What is the volume of the pyramid?
Answer Choices:
A. 392
B. 1966
C. 3922
D. 3923
E. 784 Solution:
Square ABCD has side length 14. Let F and G be the feet of the altitudes from E in △ABE and △CDE, respectively. Then FG=14, EF=2⋅14105=15 and EG=2⋅1491=13. Because △EFG is perpendicular to the plane of ABCD, the altitude to FG is the altitude of the pyramid. By Heron's Formula, the area of △EFG is (21)(6)(7)(8)=84, so the altitude to FG is 2⋅1484=12. Therefore the volume of the pyramid is (31)(196)(12)=784.