AMC8 2025

AMC8 2025 · Q24

AMC8 2025 · Q24. It mainly tests Triangles (properties), Area & perimeter.

In trapezoid $ABCD$, angles $B$ and $C$ measure $60^\circ$ and $AB = DC$. The side lengths are all positive integers, and the perimeter of $ABCD$ is 30 units. How many non-congruent trapezoids satisfy all of these conditions?

在梯形$ABCD$中，角$B$和$C$分别测得$60^\circ$且$AB = DC$。边长均为正整数，且$ABCD$的周长为30单位。满足所有这些条件的非全等的梯形有多少个？

(A) \ 0 \ 0

(B) \ 1 \ 1

(D) \ 3 \ 3

(E) \ 4 \ 4

Answer

Correct choice: (E)

正确答案：（E）

Solution

Let $a$ be the length of the shorter base, and let $b$ be the length of the longer one. Note that these two parameters, along with the angle measures and the fact that the trapezoid is isosceles, uniquely determine a trapezoid. We drop perpendiculars down from the endpoints of the top base. Then the length from the foot of this perpendicular to either vertex will be half the difference between the lengths of the two bases, or $\frac{b-a}{2}$. Now, since we have a 30-60-90 triangle and this side length corresponds to the "30" part, the length of the hypotenuse (one of the legs) is $2 \cdot \frac{b-a}{2} = b-a$. Then the perimeter of the trapezoid is $2(b-a)+a+b=3b-a=30$. The only other stipulation for this trapezoid to be valid is that $b>a$ (which was our assumption). We can now easily count the valid pairs $(a,b)$, yielding $(3,11),(6,12),(9,13),(12,14)$. It is clear that proceeding further would cause $a \geq b$, so we have $\boxed{\textbf{(E)}~4}$ valid trapezoids.

设$a$为较短底边的长度，$b$为较长底边的长度。注意这两个参数，连同角度测量和梯形是等腰的事实，唯一确定一个梯形。我们从上底端点向下垂垂直线。然后从垂足到任一顶点的长度是两条底边长度差的一半，即$\frac{b-a}{2}$。现在，由于我们有一个30-60-90三角形且这个边长对应“30”部分，斜边（一条腿）的长度是$2 \cdot \frac{b-a}{2} = b-a$。然后梯形的周长是$2(b-a)+a+b=3b-a=30$。这个梯形有效的唯一其他规定是$b>a$（这是我们的假设）。我们现在可以轻松数出有效的$(a,b)$对，即$(3,11),(6,12),(9,13),(12,14)$。显然继续下去会导致$a \geq b$，因此我们有$\boxed{\textbf{(E)}~4}$个有效梯形。

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