2009 AMC 10B Problem 12

Problem 12 of 25IntermediateGeometry Counting & Probability

Distinct points

A,

B,

C,

and

D

lie on a line, with

AB=BC=CD=1.

Points

E

and

F

lie on a second line, parallel to the first, with

EF=1.

A triangle with positive area has three of the six points as its vertices. How many possible values are there for the area of the triangle?

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Solution

A positive-area triangle uses two points on one line as its base and one point on the other line as its apex. The height is always the fixed distance between the lines, so the area depends only on the base length. Bases on the first line can be

1, 2,

3;

a base on the second line is

1.

So the distinct base lengths are

1, 2, 3,

giving three possible areas. Thus, the correct answer is A.

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