2019 AMC 12B Problem 18

Problem 18 of 25IntermediateGeometry

Square pyramid

ABCDE

has base

ABCD,

which measures

3

cm on a side, and altitude

\overline{AE}

perpendicular to the base, which measures

6

cm. Point

P

lies on

\overline{BE},

one third of the way from

B

to

E;

point

Q

lies on

\overline{DE},

one third of the way from

D

to

E;

and point

R

lies on

\overline{CE},

two thirds of the way from

C

to

E.

What is the area, in square centimeters, of

\triangle PQR?

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Solution

Place

A=(0,0,0),

\ B=(3,0,0),

\ C=(3,3,0),

\ D=(0,3,0),

\ E=(0,0,6).

Then

\begin{gathered} P=(2,0,2), \\ Q=(0,2,2), \\ R=(1,1,4). \end{gathered}

So

\vec{PQ}=(-2,2,0)

and

\vec{PR}=(-1,1,2),

giving

\vec{PQ}\times\vec{PR}=(4,4,0).

The area is

\dfrac12\bigl|(4,4,0)\bigr|=\dfrac12\cdot4\sqrt2=2\sqrt2.

Thus, C is the correct answer.

More practice

Concepts: 3D geometry · vector · triangle area

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.