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单位正方形内接等边三角形的面积是多少? – 译学馆
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单位正方形内接等边三角形的面积是多少?

Equilateral Triangle In A Square

Mind your decisions.
《谨慎决定》
I’m Presh talwalkar.
我是Presh Talwalkar
An equilateral triangle is inscribed in a unit square
一个单位正方形内切一个等边三角形
What is the area of the equilateral triangle?
那这个等边三角形的面积是多少?
Pause the video, if you’d like to give this problem a try
如果你想试着解决这个问题 就暂停视频
and when you’re ready, keep watching to
你准备好了就继续观看
learn how to solve this problem.
学习如何解决这个问题
(music)
[音乐]
如果你用头脑谨慎做出决定
那么一切都迎刃而解
So, how can we solve this problem
那么我们怎么解这题呢?
Let’s write s for the length of the equilateral triangle side.
我们用S表示等边三角形的边长
The square has a side length of 1.
正方形的边长为1
Now, in this right triangle,
现在 在这个直角三角形中
suppose the other leg is equal to X
设它的另一边为X
by the Gogu theorem s^2 is equal to 1^2 plus x^2.
根据勾股定理 S²=1²+ X²
Now consider the following right triangle
现在来看下面的直角三角形
Its hypotenuse is also equal to s and one of its legs is equal to 1
它的斜边也是S 其中一边为1
Therefore its other leg must also be equal to x
因此 它的另一边也一定是X
Finally we have one more triangle.
最后 我们还有一个三角形
It must have legs of 1 minus x
它的两条边均是1-X
because the entire length of the square side is equal to 1
因为正方形的边长是1
We therefore have s squared
因此 我们能够得到
is equal to 1 minus x the quantity squared plus 1 minus x the quantity squared
S²=(1-X)²+(1-X)²
Well simplify this a little bit and now we have two equations.
化简一下 现在我们得到两个方程式
Well consider this system of equations
看这组方程
we’ll subtract the second equation from the first
我们用第1个方程减去第2个方程
and the s squared terms will cancel out.
那么S²就消掉了
We then get an equation in one variable X
那么我们得到一个只有变量X的方程
we can routinely solve this and we get a quadratic equation.
例行解此方程 得到一个二次方程
We use the quadratic formula and then we simplify to get two possibilities
我们用二次方程式化解 然后得到两种解
x is equal to 2 minus root 3 and x is equal to 2 plus root 3
X=2-√3和X=2+√3
Now recall that X has to be less than the side of the square.
现在回想一下 X必须小于正方形的边长
So X has to be between 0 & 1
所以X的值必须在0到1之间
Therefore we reject 2 plus root 3 and we accept 2 minus root 3
因此 我们舍去2+√3 得到2-√3
once we have the value of x we can go ahead and solve for s squared and
一旦我们知道X的值 就可以继续求出S²的值
Then remember we’re trying to solve for the area of the equilateral triangle,
记住我们求的是等边三角形的面积
which is equal to s squared root 3 all over 4
那就是S²×√3除去4
we simplify this and we get 2 root 3 minus 3 ,
化简 我们得到2√3-3
which is approximately 0.464
近似值是0.464
and that’s our answer.
这就是我们的答案
Thanks for making mind your decisions one of the best channels on YouTube
感谢让《谨慎决定》 成为YouTube的最佳频道之一
as always thanks for watching and thanks for your support
一如既往地感谢您的收看与支持

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为你解答如何求正方形内接等边三角形的面积

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视频来源

https://www.youtube.com/watch?v=sFhBzffSWTE

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