Question

若三角形的一个角保持不变，而该角的两条夹边都加倍，则面积变为原来的：

If an angle of a triangle remains unchanged but each of its two including sides is doubled, then the area is multiplied by:

$2$

$3$

$4$

$6$

大于 $6$ 倍

more than $6$

答案：C

知识点：三角形面积 included angle scaling

难度评级：1180

小提示：

使用由两边及其夹角求面积的公式 $\frac12ab\sin C$

Use the area formula $\frac12ab\sin C$ for two sides and their included angle

大提示：

两条边都加倍，会使它们的乘积乘以 $2\cdot2$

Doubling both side factors multiplies their product by $2\cdot2$

解答：

若两条夹边为 $a$ 和 $b$ ，保持不变的夹角为 $C$ ，则原面积为 $\frac12ab\sin C$ 。两条边都加倍后，面积为 $\begin{gathered} \frac12(2a)(2b)\sin C \\ =2ab\sin C\\ =4\left(\frac12ab\sin C\right) \end{gathered}\text{。}$

因此，正确答案是 C。

If the included sides are $a$ and $b,$ and their unchanged angle is $C,$ the original area is $\frac12ab\sin C.$ After both sides are doubled, the area is $\begin{gathered} \frac12(2a)(2b)\sin C \\ =2ab\sin C\\ =4\left(\frac12ab\sin C\right). \end{gathered}$

Thus, the correct answer is C.

1956 AMC 12 第 27 题

答案：C

难度评级：1180

小提示：

大提示：

解答：

其他年份的第 27 题