2000 AMC 10 Problems/Problem 5

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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 $ABNM$

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

Solution

(a) Clearly $AB$ does not change, and $MN=\frac{1}{2}AB$ , so $MN$ doesn't change either.

(b) Obviously, the perimeter changes.

(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 trapezoid remains the same.

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

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

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