1.6 30-60-90 右三角关系

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  30-60-90 右三角关系

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  You are working on a project in your Industrial Arts class. You have been instructed to design a building. You have a plastic triangular piece that helps you with straight edges and designs. This triangle has interior angles of $\begin{array}{r}{30}^{\circ }\end{array}$ , $\begin{array}{r}{60}^{\circ }\end{array}$ , and $\begin{array}{r}{90}^{\circ }\end{array}$ . Laying the triangle flat on the paper, you decide to use it to help you with the right angle at the building's base.
  ::您正在从事工业艺术类的一个项目。 您已被指示要设计一座建筑。 您有一个塑料三角形板块可以帮助您使用直边和设计。 这个三角形的内角为 30 , 60 , 90 。 在纸上放置三角形的平面, 您决定用它来帮助您在建筑的基地找到正确的角度 。

  

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  You trace out the bottom and left edge of the triangle on the paper to serve as the side and bottom of the structure. Looking at the length of the base of the building, you see that it is 7 inches long. Can you determine what the height of the building is from this information?
  ::您跟踪纸上三角形的底部和左边缘, 以作为结构的侧面和底部。 查看大楼底部的长度, 您看到有7英寸长。 您能够确定此信息显示建筑物的高度吗 ?

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  30-60-90 Right Triangles
  ::30-60-90 右三角

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  $\begin{array}{r}30-60-90\end{array}$ refers to each of the angles in this special right triangle .
  ::30-60-90指这个特殊右三角形的每个角度。

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   To understand the ratios of the sides, start with an with an drawn from one vertex .
  ::为了了解两边的比例,首先从一个顶端抽取一个。

  

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  Recall from geometry that an altitude, $\begin{array}{r}h\end{array}$ , cuts the opposite side directly in half. So we know that one side, the hypotenuse , is $\begin{array}{r}2s\end{array}$ and the shortest leg is $\begin{array}{r}s\end{array}$ . Also, recall that the altitude is a perpendicular and angle bisector , which is why the angle at the top is split in half. To find the length of the longer leg, use the Pythagorean Theorem :
  ::从几何学上回顾,一个高度,h,直接将对面的两面切成两半。所以我们知道,一个侧面,即下限,是2秒,而短腿是s。还记得,高度是垂直的和角度的两部分,这就是顶部角被拆成两半的原因。要找到长腿的长度,请使用Pythagorean神话:

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  $$\begin{array}{rl}{s}^2+h^2& =(2s{)}^2\\ s^2+h^2& =4s^2\\ h^2& =3s^2\\ h& =s\sqrt{3}\end{array}$$

  
  ::s2+h2=(2s)2s2+h2=4s2=4s2h2=3s2=s3

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  From this we can conclude that the length of the longer leg is the length of the short leg multiplied by $\begin{array}{r}\sqrt{3}\end{array}$ or $\begin{array}{r}s\sqrt{3}\end{array}$ . Just like the isosceles right triangle , we now only need one side in order to determine the other two in a $\begin{array}{r}30-60-90\end{array}$ triangle. The ratio of the three sides is written $\begin{array}{r}x:x\sqrt{3}:2x\end{array}$ , where $\begin{array}{r}x\end{array}$ is the shortest leg, $\begin{array}{r}x\sqrt{3}\end{array}$ is the longer leg and $\begin{array}{r}2x\end{array}$ is the hypotenuse.
  ::由此我们可以得出结论,长腿的长度是短腿的长度乘以3或S3。 正如右三角的等骨一样,我们现在只需要一面来确定30-60-90三角的另外两边。 三个边的比是写成的 x:x3:2x, 其中x是短腿,x3是长腿,2x是低脚。

  

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  Notice, that the shortest side is always opposite the smallest angle and the longest side is always opposite $\begin{array}{r}{90}^{\circ }\end{array}$ .
  ::注意,最短的一面总是对着最小的一角,最长的一面总是对着90。

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  Finding the Length of Missing Sides 
  ::查找缺失侧的长度

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  1. Find the lengths of the two missing sides in the $\begin{array}{r}30-60-90\end{array}$ triangle.
  ::1. 找出30-60-90三角地带两个失踪方的长度。

  

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   Determine which side in the $\begin{array}{r}30-60-90\end{array}$ ratio is given and solve for the other two.
  ::确定30-60-90比率的哪一边,并解决另外两个方面。

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  $\begin{array}{r}4\sqrt{3}\end{array}$ is the longer leg because it is opposite the $\begin{array}{r}{60}^{\circ }\end{array}$ . So, in the $\begin{array}{r}x:x\sqrt{3}:2x\end{array}$ ratio, $\begin{array}{r}4\sqrt{3}=x\sqrt{3}\end{array}$ , therefore $\begin{array}{r}x=4\end{array}$ and $\begin{array}{r}2x=8\end{array}$ . The short leg is 4 and the hypotenuse is 8.
  ::43是长腿,因为它与60°Q相对。因此,在x:x3:2x比率、43=x3、因此x=4和2x=8中,短腿为4, 下限为8。

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  2. Find the lengths of the two missing sides in the $\begin{array}{r}30-60-90\end{array}$ triangle.
  ::2. 找出30-60-90三角地带两个失踪方的长度。

  

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   Determine which side in the $\begin{array}{r}30-60-90\end{array}$ ratio is given and solve for the other two.
  ::确定30-60-90比率的哪一边,并解决另外两个方面。

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  17 is the hypotenuse because it is opposite the right angle. In the $\begin{array}{r}x:x\sqrt{3}:2x\end{array}$ ratio, $\begin{array}{r}17=2x\end{array}$ and so the short leg is $\begin{array}{r}\frac{17}{2}\end{array}$ and the long leg is $\begin{array}{r}\frac{17\sqrt{3}}{2}\end{array}$ .
  ::17 是右角对面的下限值。 在 x:x3:2x比率中, 17=2x, 所以短腿是172, 长腿是1732 。

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  3. Find the lengths of the two missing sides in the $\begin{array}{r}30-60-90\end{array}$ triangle.
  ::3. 找出30-60-90三角地带两个失踪方的长度。

  

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   Determine which side in the $\begin{array}{r}30-60-90\end{array}$ ratio is given and solve for the other two.
  ::确定30-60-90比率的哪一边,并解决另外两个方面。

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  15 is the long leg because it is opposite the $\begin{array}{r}{60}^{\circ }\end{array}$ . Even though 15 does not have a radical after it, we can still set it equal to $\begin{array}{r}x\sqrt{3}\end{array}$ .
  ::15是长腿,因为它对面是60。即使15没有激进的后腿,我们仍然可以将其设为x3。

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  $$\begin{array}{rl}x\sqrt{3}& =15\\ x& =\frac{15}{\sqrt{3}}\cdot \frac{\sqrt{3}}{\sqrt{3}}=\frac{15\sqrt{3}}{3}=5\sqrt{3}\text{So, the short leg is}5\sqrt{3}.\end{array}$$

  
  ::x3=15x=1533=15333=15333=53So,短腿为53。

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  Multiplying $\begin{array}{r}5\sqrt{3}\end{array}$ by 2, we get the hypotenuse length, which is $\begin{array}{r}10\sqrt{3}\end{array}$ .
  ::乘以53乘以53乘以2 我们得到的时空长度是103

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were asked to design a building. 
  ::早些时候,你被要求设计一栋建筑。

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  Since you know the ratios of sides of a 30-60-90 triangle , you know that since the bottom side has a length of $\begin{array}{r}7\end{array}$ cm, the left side must have a length of $\begin{array}{r}7\sqrt{3}\approx 12.12\end{array}$ cm.
  ::既然你知道30-60-90三角形两侧的比率, 你知道,既然底侧长度为7厘米, 左侧长度必须7312.12厘米。

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  Example 2
  ::例2

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  Find the lengths of the two missing sides in the $\begin{array}{r}30-60-90\end{array}$ triangle.
  ::查找30-60-90三角形中两个失踪方的长度。

  

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   Since the given side has a length of 8 and is the side which is opposite the right angle, we know that this is the "2x" side of the triangle. Therefore, the short side of the triangle is $\begin{array}{r}\frac{8}{2}=4\end{array}$ and the third side of the triangle is $\begin{array}{r}4\sqrt{3}\end{array}$ .
  ::由于给定方的长度为8, 是右角对面的侧, 我们知道这是三角形的“ 2x ” 侧面。 因此, 三角形的短边是 82= 4, 三角形的第三边是 43 。

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  Example 3
  ::例3

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  Find the lengths of the two missing sides in the 30-60-90 triangle below.
  ::在下面的30-60-90三角形中 找到两个失踪方的长度

  

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  We can see that the shortest side of the triangle has a length of 3 while the longest side has a length of 6 and the other side has a length of $\begin{array}{r}3\sqrt{3}\end{array}$ . This means that the $\begin{array}{r}{30}^{\circ }\end{array}$ angle is opposite the side with length $\begin{array}{r}3\end{array}$ , the $\begin{array}{r}{60}^{\circ }\end{array}$ angle is opposite the side with length $\begin{array}{r}3\sqrt{3}\end{array}$ , and the $\begin{array}{r}{90}^{\circ }\end{array}$ angle is opposite the side with length $\begin{array}{r}6\end{array}$ .
  ::可以看到三角形最短的侧面长为3,而最长的侧面长为6,而另一边长为33。 这意味着30角对面,长度为3,60角对面,长度为33,90角对面,长度为6。

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  Example 4
  ::例4

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  Find the angles in the $\begin{array}{r}30-60-90\end{array}$ triangle.
  ::在 30- 60- 90 三角形中查找角度 。

  

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  The length of the given side is $\begin{array}{r}3\sqrt{3}\end{array}$ , and is opposite the $\begin{array}{r}{60}^{\circ }\end{array}$ angle. This means that the side opposite the $\begin{array}{r}{30}^{\circ }\end{array}$ angle is $\begin{array}{r}3\end{array}$ , and the length of the side opposite the $\begin{array}{r}{90}^{\circ }\end{array}$ angle is $\begin{array}{r}6\end{array}$
  ::给定方的长度为 33, 与 60 角度相反 。 这意味着 30 角度对面的长度为 3 , 90 角度对面的长度为 6 。

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  Review
  ::回顾

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  1. In a 30-60-90 triangle, if the shorter leg is 8, then the longer leg is __________ and the hypotenuse is ___________.
     ::在30-60-90三角形中,如果短腿为8,则长腿为________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
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  2. In a 30-60-90 triangle, if the shorter leg is 12, then the longer leg is __________ and the hypotenuse is ___________.
     ::在30-60-90三角形中,如果短腿为12,则长腿为____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
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  3. In a 30-60-90 triangle, if the longer leg is 10, then the shorter leg is __________ and the hypotenuse is ___________.
     ::在30-60-90三角形中,如果长腿为10,那么短腿为____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
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  4. In a 30-60-90 triangle, if the shorter leg is 16, then the longer leg is __________ and the hypotenuse is ___________.
     ::在30-60-90三角形中,如果短腿为16,则长腿为____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
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  5. In a 30-60-90 triangle, if the longer leg is 3, then the shorter leg is __________ and the hypotenuse is ___________.
     ::在30-60-90三角形中,如果长腿为3,那么短腿为____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
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  6. In a 30-60-90 triangle, if the shorter leg is $\begin{array}{r}x\end{array}$ , then the longer leg is __________ and the hypotenuse is ___________.
     ::在30-60-90三角形中,如果短腿为x,则长腿为,下限为。
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  7. In a 30-60-90 triangle, if the longer leg is $\begin{array}{r}x\end{array}$ , then the shorter leg is __________ and the hypotenuse is ___________.
     ::在30-60-90三角形中,如果长腿是x,那么短腿是,下限是。
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  8. A rectangle has sides of length 7 and $\begin{array}{r}7\sqrt{3}\end{array}$ . What is the length of the diagonal?
     ::矩形有7和73的两边,对角线的长度是多少?
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  9. Two (opposite) sides of a rectangle are 15 and the diagonal is 30. What is the length of the other two sides?
     ::矩形的两面(对面)是15,对角是30,其他两面的长度是多少?
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  10. What is the height of an equilateral triangle with sides of length 6 in?
      ::宽度为 6 的两边的等边三角形高度是多少 ?
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  11. What is the area of an equilateral triangle with sides of length 10 ft?
      ::长10英尺长的等边三角形的面积是多少?
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  12. A regular hexagon has sides of length 3 in. What is the area of the hexagon?
      ::普通六边形的长度为3。六边形的面积是多少?
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  13. The area of an equilateral triangle is $\begin{array}{r}36\sqrt{3}\end{array}$ . What is the length of a side?
      ::等边三角形的区域是 363 。 侧边的长度是多少 ?
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  14. If a road has a grade of $\begin{array}{r}{30}^{\circ }\end{array}$ , this means that its angle of elevation is $\begin{array}{r}{30}^{\circ }\end{array}$ . If you travel 3 miles on this road, how much elevation have you gained in feet (5280 ft = 1 mile)?
      ::如果一条道路的等级为30英寸,这意味着其高度角度是30英寸。如果你在这条道路上行驶3英里,你脚上升了多少英尺(5280英尺=1英里)?
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  15. If a road has a grade of $\begin{array}{r}{30}^{\circ }\end{array}$ , this means that its angle of elevation is $\begin{array}{r}{30}^{\circ }\end{array}$ . If you travel x miles on this road, how much elevation have you gained in feet (5280 ft = 1 mile)?
      ::如果一条公路的等级为30英寸,这意味着其高度角度是30英寸。如果你在这条公路上行驶x英里,你脚上升了多少英尺(5280英尺=1英里)?

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  Review (Answers)
  ::回顾(答复)

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  Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
  ::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键 。