2017 AMC 12A Problem 9

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?

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Solution

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).

3=y-4\ge x+2:

then

y=7

and

x\le1,

a leftward ray from

(1,7).

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),

S

is three rays with a common endpoint. Thus, the correct answer is E.

Tagged: coordinate geometry · casework