2020 AMC 10B Problem 8

Question

Points

P

and

Q

lie in a plane with

PQ=8.

How many locations for point

R

in this plane are there such that the triangle with vertices

P,

Q,

and

R

is a right triangle with area

12

square units?

2

4

6

8

12

Answer: D

Concepts:right triangle triangle area coordinate geometry casework

Difficulty rating: 1420

Small hint:

The area condition fixes the distance from

R

to line

PQ

Big hint:

Separate the cases where the right angle is at

P,

Q,

or

R

Solution:

Place

P=(-4,0)

and

Q=(4,0).

Since the area is

12

and

PQ=8,

the distance from

R

to line

PQ

is

3,

so

R=(x,\pm 3).

If the right angle is at

P,

then

R=(-4,\pm 3),

giving

2

points. If it is at

Q,

then

R=(4,\pm 3),

giving

2

more points.

If the right angle is at

R,

then

PQ

is the hypotenuse, so

\begin{aligned} &64=PR^2+QR^2 \\ &\quad =(x+4)^2+9 \\ &\quad {}+(x-4)^2+9 \\ &\quad =2x^2+50. \end{aligned}

Thus

x^2=7,

giving

R=(\pm\sqrt7,\pm 3),

another

4

points.

The total is

2+2+4=8.

Thus, the correct answer is D .

2020 AMC 10B Problem 8

Answer: D

Difficulty rating: 1420

Small hint:

Big hint:

Solution:

Problem 8 in other years