题目内容
如图甲,把正方形ACFG和Rt△ACB按如图甲所示重叠在一起,其中AC=2,∠BAC=60°,若把Rt△ACB绕直角顶点C按顺时针方向旋转,使斜边AB恰好经过正方形的顶点F,得△A′B′C,AB分别与A′C,A′B′相交于D,E,如图乙所示,那么△ACB与△A′B′C的重叠部分(即阴影部分)的面积为 .![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021230723832307372/SYS201310212307238323073015_ST/images0.png)
【答案】分析:根据题意,建立坐标系,易得A′C′、AB、A′B′的直线方程,进而可得E、H的坐标,计算可得S△A′DE,由图形间的关系,计算可得答案.
解答:
解:如图取坐标系,易知,△A′CF为正三角形,A′C方程:y=
x①
AB方程:
+
=1②,A′B′方程:y=-
(x-2)③.
①②得D(
,
). ②③得E(3-
,3-
).
又A′(1,
)
y=3-
,与①联立得H(2-
,3-
),
高=2
-3,底=2-
.
S△A′DE=[(2
-3)(2-
)]÷2=
-6,
SAFED=S△A′CF-S△A′DE=6-
.
点评:本题考查面积的计算,涉及旋转的性质与运用,注意结合函数的基本性质解题.
解答:
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021230723832307372/SYS201310212307238323073015_DA/images0.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021230723832307372/SYS201310212307238323073015_DA/0.png)
AB方程:
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021230723832307372/SYS201310212307238323073015_DA/1.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021230723832307372/SYS201310212307238323073015_DA/2.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021230723832307372/SYS201310212307238323073015_DA/3.png)
①②得D(
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021230723832307372/SYS201310212307238323073015_DA/4.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021230723832307372/SYS201310212307238323073015_DA/5.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021230723832307372/SYS201310212307238323073015_DA/6.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021230723832307372/SYS201310212307238323073015_DA/7.png)
又A′(1,
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021230723832307372/SYS201310212307238323073015_DA/8.png)
y=3-
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021230723832307372/SYS201310212307238323073015_DA/9.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021230723832307372/SYS201310212307238323073015_DA/10.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021230723832307372/SYS201310212307238323073015_DA/11.png)
高=2
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021230723832307372/SYS201310212307238323073015_DA/12.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021230723832307372/SYS201310212307238323073015_DA/13.png)
S△A′DE=[(2
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021230723832307372/SYS201310212307238323073015_DA/14.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021230723832307372/SYS201310212307238323073015_DA/15.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021230723832307372/SYS201310212307238323073015_DA/16.png)
SAFED=S△A′CF-S△A′DE=6-
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021230723832307372/SYS201310212307238323073015_DA/17.png)
点评:本题考查面积的计算,涉及旋转的性质与运用,注意结合函数的基本性质解题.
![](http://thumb2018.1010pic.com/images/loading.gif)
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