AMC8 2026

AMC8 2026 · Q3

AMC8 2026 · Q3. It mainly tests Triangles (properties), Pythagorean theorem.

Haruki has a piece of wire that is $24$ centimeters long. He wants to bend it to form each of the following shapes, one at a time: A regular hexagon with side length $5$ cm. A square with area $36 \hspace{3pt} \text{cm}^2$. A right triangle whose legs are $6$ and $8$ cm long. Which of the shapes can Haruki make?

春树有一段长为 $24$ 厘米的铁丝。他想将它弯成下列每一种形状，依次折成：边长为 $5$ 厘米的正六边形。面积为 $36 \hspace{3pt} \text{cm}^2$ 的正方形。两条直角边分别为 $6$ 厘米和 $8$ 厘米的直角三角形。春树能做出哪几种形状？

(A) \text{Triangle only} \hspace{3pt} \text{Triangle only} \\

(B) \text{Hexagon and square only} \hspace{3pt} \text{Hexagon and square only} \\

(D) \text{Square and triangle only} \hspace{3pt} \text{Square and triangle only} \\

(E) \text{Hexagon, triangle, and square} \hspace{3pt} \text{Hexagon, triangle, and square}

Answer

Correct choice: (D)

正确答案：（D）

Solution

To solve this problem we can find the perimeter of each of the given spaces and check if it is less than 24. The first shape is a regular hexagon with side length $5$. The hexagon has a perimeter of $6 \cdot 5 = 30$ centimeters. The second shape is a square with area $36$ or a side length of $6$. Then the perimeter of all $4$ sides is $24$ centimeters. Finally, we need to find the perimeter of a right triangle with legs $6$ and $8$ centimeters. The hypotenuse of the right triangle is $10$ so the total perimeter is $6 + 8 + 10 = 24$ centimeters. Thus, only the square and the triangle can be made with 24 centimeters of wire so the answer is $\boxed{D}$.

为了解决这个问题，我们可以计算每种给定形状的周长，并检查其是否小于24厘米。第一个形状是边长为 $5$ 的正六边形，周长为 $6 \cdot 5 = 30$ 厘米。第二个形状是面积为 $36$ 的正方形，其边长为 $6$ 厘米，则四条边的周长为 $24$ 厘米。最后，需要求两条直角边分别为 $6$ 和 $8$ 厘米的直角三角形的周长。该直角三角形的斜边为 $10$ 厘米，因此总周长为 $6 + 8 + 10 = 24$ 厘米。因此，只有正方形和三角形可以用 $24$ 厘米的铁丝制作，答案是 $\boxed{D}$。

Topics