2000 AMC 10 Problems/Problem 5

Problem

Points $M$ and $N$ are the midpoints of sides $PA$ and $PB$ of $\triangle PAB$ . As $P$ moves along a line that is parallel to side $AB$ , how many of the four quantities listed below change?

(a) the length of the segment $MN$

(b) the perimeter of $\triangle PAB$

(c) the area of $\triangle PAB$

(d) the area of trapezoid $ABNM$

$[asy] draw((2,0)--(8,0)--(6,4)--cycle); draw((4,2)--(7,2)); draw((1,4)--(9,4),Arrows); label("$A$",(2,0),SW); label("$B$",(8,0),SE); label("$M$",(4,2),W); label("$N$",(7,2),E); label("$P$",(6,4),N); [/asy]$

$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4$

Solution

(a) Triangles $ABP$ and $MNP$ are similar, and since $PM=\frac{1}{2}AP$ , $MN=\frac{1}{2}AB$ .

(b) Obviously, the perimeter changes. For example, imagine if P was extremely far to the left.

(c) The area clearly doesn't change, as both the base $AB$ and its corresponding height remain the same.

(d) The bases $AB$ and $MN$ do not change, and neither does the height, so the area of the trapezoid remains the same.

Only $1$ quantity changes, so the correct answer is $\boxed{\text{B}}$ .

2000 AMC 10 (Problems • Answer Key • Resources)
Preceded by Problem 4	Followed by Problem 6
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2000 AMC 10 Problems/Problem 5

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