Question

Jean 做了一件彩色玻璃艺术品，形状像两座山，如下图所示。一座山峰高 $8$ 英尺，另一座山峰高 $12$ 英尺。每个山峰形成一个 $90^\circ$ 角，山的直边与地面成 $45^\circ$ 角。艺术品面积为 $183$ 平方英尺。两座山的边在艺术品中心附近相交，交点离地面 $h$ 英尺。 $h$ 的值是多少？

Jean made a piece of stained glass art in the shape of two mountains, as shown in the figure below. One mountain peak is $8$ feet high and the other peak is $12$ feet high. Each peak forms a $90^\circ$ angle, and the straight sides of the mountains form $45^\circ$ angles with the ground. The artwork has an area of $183$ square feet. The sides of the mountains meet at an intersection point near the center of the artwork, $h$ feet above the ground. What is the value of $h?$

$4$

$5$

$4\sqrt{2}$

$6$

$5\sqrt{2}$

答案：B

知识点：特殊直角三角形面积容斥原理

难度评级：1710

视频讲解：

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文字解答：

每座山都是 $45^\circ\text{-}45^\circ\text{-}90^\circ$ 三角形。高为 $x$ 的这种直角等腰三角形面积为 $x^2$ ，因为两条直角边都长 $x\sqrt2$ 。

两座大山的面积分别为 $8^2=64$ 和 $12^2=144$ 。重叠部分也是 $45^\circ\text{-}45^\circ\text{-}90^\circ$ 三角形，高为 $h$ ，面积为 $h^2$ 。因此得 $h^2=25$ ，所以 $h=5$ 。 $64+144-h^2=183,$

正确答案是 B。

Each mountain is a $45^\circ\text{-}45^\circ\text{-}90^\circ$ triangle. A right isosceles triangle with height $x$ has area $x^2$ , since its two perpendicular sides each have length $x\sqrt2$ .

The two large mountains have areas $8^2=64$ and $12^2=144$ . Their overlap is also a $45^\circ\text{-}45^\circ\text{-}90^\circ$ triangle with height $h$ , so its area is $h^2$ . Thus $64+144-h^2=183,$ so $h^2=25$ , and $h=5$ .

Thus, B is the correct answer.

2024 AMC 8 第 24 题

答案：B

难度评级：1710

视频讲解：

文字解答：

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