1984 AMC 12 Problem 10

Attempt Problem 10 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.

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10.

Four complex numbers lie at the vertices of a square in the complex plane. Three of the numbers are

1+2i,

-2+i

and

-1-2i.

The fourth number is

2+i

2-i

1-2i

-1+2i

-2-i

Two of the listed vertices are opposites

Their midpoint is the square’s center, so reflect the remaining given vertex through that point

The points

1+2i

and

-1-2i

are opposite vertices, so their midpoint

0

is the center of the square. The fourth vertex is therefore the reflection of

-2+i

through the origin, namely

2-i.

Therefore, the correct answer is B.

Problem 10 in other years