2017 AMC 12A Problem 9

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

Set two of the three quantities equal and require the third to be at most that common value

Each of the three pairings produces a ray, and all three start at the same point

(1,7)

Consider which two of

3,

x+2,

y-4

are the (equal) larger pair.

If

3=x+2\ge y-4:

then

x=1

and

y\le7,

a downward ray from

(1,7).

If

3=y-4\ge x+2:

then

y=7

and

x\le1,

a leftward ray from

(1,7).

If

x+2=y-4\ge3:

then

y=x+6

and

x\ge1,

a ray from

(1,7)

going up and to the right.

All three rays share the endpoint

(1,7),

so

S

is three rays with a common endpoint.

Thus, the correct answer is E.

Problem 9 in other years