2007 AMC 10B Problem 10

Below is the professionally curated solution for Problem 10 of the 2007 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2007 AMC 10B solutions, or check the answer key.

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Concepts:triangle area parallel lines

Difficulty rating: 1270

10.

Two points $B$ and $C$ are in a plane. Let $S$ be the set of all points $A$ in the plane for which $\triangle ABC$ has area $1.$ Which of the following describes $S?$

two parallel lines

a parabola

a circle

a line segment

two points

Solution:

Taking $BC$ as the base, the area is $\dfrac12(BC)d,$ where $d$ is the distance from $A$ to line $BC.$ The area equals $1$ exactly when $d=\dfrac{2}{BC}.$

The points at this fixed distance from line $BC$ form two lines parallel to $BC,$ one on each side.

Thus, the correct answer is A.

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