2007 AMC 10B Problems/Problem 10

Problem

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 $S?$

$\textbf{(A) } \text{two parallel lines} \qquad\textbf{(B) } \text{a parabola} \qquad\textbf{(C) } \text{a circle} \qquad\textbf{(D) } \text{a line segment} \qquad\textbf{(E) } \text{two points}$

Solution

Let $h$ be the length of the altitude of $\triangle ABC.$ Since segment $BC$ is the base of the triangle and cannot change, the area of the triangle is $\frac{1}{2}(BC)(h)=1$ and $h=\frac{2}{BC}.$ Thus $S$ consists of two lines parallel to $BC$ and is $\frac{2}{BC}$ units away from it. $\longrightarrow \boxed{\mathrm{(A) \ } \text{two parallel lines}}$

See Also

2007 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 9
Followed by
Problem 11
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All AMC 10 Problems and Solutions

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