Question

等腰三角形是有两条边长度相等的三角形。下面方格图中的五个三角形中，有多少个是等腰三角形？

An isosceles triangle is a triangle with two sides of equal length. How many of the five triangles on the square grid below are isosceles?

$1$

$2$

$3$

$4$

$5$

答案：D

知识点：等腰三角形距离公式

难度评级：960

解答：

从方格中读出每个三角形的边长。水平或竖直边可直接数格子；一条斜边横跨 $a$ 个单位、竖跨 $b$ 个单位，长度为 $\sqrt{a^2 + b^2}$ 。

左上三角形：两条斜边都横跨 $1$ 、竖跨 $2$ ，所以长度都为 $\sqrt{5}$ ，底边长 $2$ 。有两边相等，所以是等腰三角形。

上方中间三角形：它有一条长 $2$ 的竖边和一条长 $2$ 的横边（第三边是斜边 $2\sqrt{2}$ ）。有两边相等，所以是等腰三角形。

右上三角形：边长是 $2$ 、 $\sqrt{5}$ 和 $\sqrt{13}$ ，三者都不同，所以不是等腰三角形。

左下（宽而扁的）三角形：两条斜边都横跨 $3$ 、竖跨 $1$ ，所以长度都为 $\sqrt{10}$ 。有两边相等，所以是等腰三角形。

右下三角形：其中两条边都横跨 $2$ 和 $1$ （长度为 $\sqrt{5}$ ），第三边为 $\sqrt{10}$ 。有两边相等，所以是等腰三角形。

五个三角形中有四个是等腰三角形。

所以正确答案是 D。

Read each triangle's side lengths from the grid. A horizontal or vertical side is counted directly, and a slanted side spanning $a$ units across and $b$ units up has length $\sqrt{a^2 + b^2}.$

Top-left triangle: its two slanted sides each span $1$ across and $2$ up, so each has length $\sqrt{5},$ with a base of length $2.$ Two equal sides, so it is isosceles.

Top-middle triangle: it has a vertical side of length $2$ and a horizontal side of length $2$ (its third side is the slant $2\sqrt{2}$ ). Two equal sides, so it is isosceles.

Top-right triangle: its sides are $2,$ $\sqrt{5},$ and $\sqrt{13},$ which are all different, so it is not isosceles.

Bottom-left (wide, flat) triangle: its two slanted sides each span $3$ across and $1$ up, so each has length $\sqrt{10}.$ Two equal sides, so it is isosceles.

Bottom-right triangle: two of its sides each span $2$ and $1$ (giving length $\sqrt{5}$ ), while the third is $\sqrt{10}.$ Two equal sides, so it is isosceles.

Four of the five triangles are isosceles.

Thus, the correct answer is D .

1988 AMC 8 第 9 题

答案：D

难度评级：960

解答：

其他年份的第 9 题