AMC 8 2024 Problem 24

AMC8 2024 年第 24 题

面积与周长（综合）★★★☆☆

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° angle, and the straight sides of the mountains form 45° 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?

Jean制作了一件两座山形状的彩色玻璃艺术品，如下图所示。一座山峰高8英尺，另一座山峰高12英尺。每座山的顶部都形成一个90°的角，两座山的直边都与地面形成一个45°的夹角。这件艺术品的面积为183平方英尺。两座山的另一边在艺术品中心附近的位置相交，相交点距离地面h英尺。请问h的值是多少？

A.4
B.5
C. $4\sqrt{2}$
D.6
E. $5\sqrt{2}$

正确答案 · Correct Answer

解析 · Solution

解法1

每座山是顶角

90^\circ

、底角

45^\circ

的等腰直角三角形。这种三角形若高为

a

，底就是

2a

，面积

=\tfrac12\cdot 2a\cdot a=a^2

。所以两座山面积分别为

8^2=64

、

12^2=144

。两山在离地

h

处交叠，重叠部分也是一个高为

h

的等腰直角三角形，面积

h^2

。整幅作品（两山的并集）面积

=64+144-h^2=183

，解得

h^2=25

，故 h=5。

Solution 1

Each mountain is a 45–45–90 (isosceles right) triangle with the right angle at the peak. Such a triangle of height

a

has base

2a

and area

\tfrac12\cdot 2a\cdot a=a^2

, so the two mountains have areas

8^2=64

and

12^2=144

. Where they overlap,

h

feet up, the doubly-covered region is another isosceles right triangle of height

h

and area

h^2

. The artwork (the union of the two mountains) has area

64+144-h^2=183

, so

h^2=25

and h=5.

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