Question

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?

$3$

$4$

$5$

$6$

$7$

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 $1, 2,$ or $3;$ 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.

2009 AMC 10B Problem 12

Solution:

Problem 12 in Other Years