2009 AMC 8 第 19 题

Question

一个等腰三角形的两个角分别为

70^\circ

和

x^\circ

。

x

的三个可能值之和是多少？

Two angles of an isosceles triangle measure

70^\circ

and

x^\circ.

What is the sum of the three possible values of

x?

95

125

140

165

180

已知的

70^\circ

角和

x^\circ

角在等腰三角形中有三种可能位置

There are three ways the known

70^\circ

angle and the

x^\circ

angle can sit in an isosceles triangle.

分别考虑它们是否为相等的角，或者其中一个是否为顶角

Consider whether they are the equal angles, or whether one is the vertex angle.

所有可能情形如下图所示。

在第一种情形中，由等腰三角形性质可得

x = 70

。

在第二种情形中，有

70 \cdot 2 + x = 180\text{，}

因此

x = 40

。

在第三种情形中，有

2x + 70 = 180\text{，}

因此

x = 55

。

这些值之和为

70 + 40 + 55 = 165\text{。}

所以正确答案是 D。

All the following possibilities are shown below.

In the first scenario, we get

x = 70

by the properties of the isosceles triangle.

In the second scenario, we get that

70 \cdot 2 + x = 180,

from which we get that

x = 40.

From the third scenario, we get that

2x + 70 = 180,

from which we get that

x = 55.

The sum of these values yields

70 + 40 + 55 = 165.

Thus, D is the correct answer.

其他年份的第 19 题