2016 AMC 10A Problem 21

Question

Circles with centers

P,

Q

and

R,

having radii

1,

2

and

3,

respectively, lie on the same side of line

l

and are tangent to

l

at

P',

Q'

and

R',

respectively, with

Q'

between

P'

and

R'.

The circle with center

Q

is externally tangent to each of the other two circles. What is the area of

\triangle PQR?

0

\sqrt{\dfrac{2}{3}}

1

\sqrt{6}-\sqrt{2}

\sqrt{\dfrac{3}{2}}

Use horizontal distances between tangent points

Use those horizontal distances in a coordinate-area formula

Put the tangent line on the

x

-axis and take

P=(0,1).

Because

PQ=1+2=3

and the centers differ in height by

1,

the horizontal distance from

P

to

Q

is

\sqrt{3^2-1^2}=2\sqrt2.

Similarly,

QR=2+3=5

and its centers also differ in height by

1,

so their horizontal distance is

2\sqrt6.

Thus we may use

\begin{aligned} P&=(0,1),\qquad Q=(2\sqrt2,2), \\ R&=(2\sqrt2+2\sqrt6,3). \end{aligned}

The coordinate-area formula gives

\begin{aligned} [PQR] &=\frac12\left|4\sqrt2-(2\sqrt2+2\sqrt6)\right| \\ &=\sqrt6-\sqrt2. \end{aligned}

Thus, the correct answer is D .

Problem 21 in other years