2008 AMC 12B Problem 18

Problem 18 of 25IntermediateGeometry

A pyramid has a square base

ABCD

and vertex

E.

The area of square

ABCD

196,

and the areas of

\triangle ABE

and

\triangle CDE

are

105

and

91,

respectively. What is the volume of the pyramid?

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Solution

The square has side

\sqrt{196} = 14.

Let

F

and

G

be the feet of the perpendiculars from

E

AB

and

CD.

Then

FG = 14,

EF = \tfrac{2 \cdot 105}{14} = 15,

and

EG = \tfrac{2 \cdot 91}{14} = 13.

Triangle

EFG

lies in a plane perpendicular to the base, so its altitude to

FG

is the pyramid’s height. By Heron’s formula with

s = 21,

its area is

\sqrt{21 \cdot 6 \cdot 8 \cdot 7} = 84,

so the altitude to

FG

\tfrac{2 \cdot 84}{14} = 12.

The volume is

\tfrac13 \cdot 196 \cdot 12 = 784.

Thus, the correct answer is E.