2008 AMC 12B Problem 18

Question

A pyramid has a square base

ABCD

and vertex

E.

The area of square

ABCD

is

196,

and the areas of

\triangle ABE

and

\triangle CDE

are

105

and

91,

respectively. What is the volume of the pyramid?

392

196\sqrt{6}

392\sqrt{2}

392\sqrt{3}

784

The base has side

14;

the given triangle areas give slant distances

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

and

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

to sides

AB

and

CD

Triangle

EFG

has sides

15, 13, 14

and stands perpendicular to the base; its altitude to

FG = 14

is the pyramid’s height

The square has side

\sqrt{196} = 14.

Let

F

and

G

be the feet of the perpendiculars from

E

to

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

is

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

The volume is

\tfrac13 \cdot 196 \cdot 12 = 784.

Thus, the correct answer is E.

Problem 18 in other years