AMC10 2000 A

AMC10 2000 A · Q5

AMC10 2000 A · Q5. It mainly tests Triangles (properties), Similarity.

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 $ABNM$

点$M$和$N$分别是$ riangle PAB$的边$PA$和$PB$的中点。当$P$沿平行于边$AB$的直线移动时，下列四个量中有多少个会变化？(a) 线段$MN$的长度 (b) $\triangle PAB$的周长 (c) $\triangle PAB$的面积 (d) 梯形$ABNM$的面积

(A) 0 0

(B) 1 1

(D) 3 3

(E) 4 4

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): Since $\triangle ABP$ is similar to $\triangle MNP$ and $PM=\frac{1}{2}\cdot AP$, it follows that $MN=\frac{1}{2}\cdot AB$. Since the base $AB$ and the altitude to $AB$ of $\triangle ABP$ do not change, the area does not change. The altitude of the trapezoid is half that of the triangle, and the bases do not change as $P$ changes, so the area of the trapezoid does not change. Only the perimeter changes (reaching a minimum when $\triangle ABP$ is isosceles).

答案（B）：由于 $\triangle ABP$ 与 $\triangle MNP$ 相似，且 $PM=\frac{1}{2}\cdot AP$，因此 $MN=\frac{1}{2}\cdot AB$。由于 $\triangle ABP$ 的底边 $AB$ 以及对应于 $AB$ 的高不变，所以面积不变。梯形的高是三角形高的一半，并且当点 $P$ 变化时两条底边不变，因此梯形的面积也不变。只有周长会变化（当 $\triangle ABP$ 为等腰三角形时周长达到最小值）。

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