1984 AMC 12 Problem 18

Attempt Problem 18 of the 1984 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1984 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

18.

A point

(x,y)

is to be chosen in the coordinate plane so that it is equally distant from the

x

-axis, the

y

-axis, and the line

x+y=2.

Then

x

\sqrt2-1

\frac12

2-\sqrt2

1

not uniquely determined

Equal distance from the two axes forces

|x|=|y|

Check both lines

y=x

and

y=-x

Equal distance from the axes gives

y=x

y=-x.

y=-x,

the distance to the line

x+y=2

\sqrt2,

so both

(\sqrt2,-\sqrt2)

and

(-\sqrt2,\sqrt2)

satisfy all three distance conditions. Their

x

-coordinates differ, so

x

is not uniquely determined.

Therefore, the correct answer is E.

Problem 18 in other years