2024 AMC 8 第 24 题

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

小提示：

每座山都由一个

45^\circ

-

45^\circ

-

90^\circ

三角形构成

Each mountain is built from a

45^\circ

-

45^\circ

-

90^\circ

triangle

大提示：

把两个大三角形面积相加，再减去重叠部分

Add the two large triangle areas and subtract the overlap

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

64+144-h^2=183\text{，}

所以

h^2=25

，进而

h=5

。

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.