2017 AMC 10A Problem 12

Question

Let

S

be the set of points

(x,y)

in the coordinate plane such that two of the three quantities

3,

x+2,

and

y-4

are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description of

S?

a single point

two intersecting lines

three lines whose pairwise intersections are three distinct points

a triangle

three rays with a common endpoint

Answer: E

Concepts:coordinate geometry casework

Difficulty rating: 1540

Small hint:

Split into cases according to which two quantities are equal

Big hint:

Each equality gives a ray once the third quantity is bounded above

Solution:

Let us case on which of the values are equal. If

3 = x + 2,

then

x = 1.

This also tells us that

y - 4 \leq 3

y \leq 7.

This describes a ray starting at

(1, 7)

and extending in the negative

y

direction.

Similarly, if

3 = y - 4,

then

y = 7

and

x + 2 \leq 3

x \leq 1.

This also describes a ray starting at

(1, 7)

but instead extending in the negative

x

direction.

Finally, if

x + 2 = y - 4,

then we have the line

y = x + 6.

Furthermore, we have that

3 \leq x + 2

x \geq 1

and

3 \leq y - 4

y \geq 7.

Note that if one of these conditions is met, the other is also necessarily true due to the equation of the line.

If

y = 7,

then

x = 1.

The other points are along the line, where

y \gt 7

and

x \gt 1.

This describes another ray that starts at

(1, 7)

and goes off in some third direction.

All three cases result in rays originating from

(1, 7)

that all go in different directions.

Thus, E is the correct answer.

2017 AMC 10A Problem 12

Answer: E

Difficulty rating: 1540

Small hint:

Big hint:

Solution:

Problem 12 in other years