5.7 三角不平等理论

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  三角不平等理论

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  Triangle Inequality Theorem
  ::三角不平等理论

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  Can any three lengths make a triangle ? The answer is no. For example, the lengths 1, 2, 3 cannot make a triangle because $\begin{array}{r}1+2=3\end{array}$ , so they would all lie on the same line . The lengths 4, 5, 10 also cannot make a triangle because $\begin{array}{r}4+5=9<10\end{array}$ . Look at the pictures below:
  ::任何三个长度能设定三角形吗? 答案是 否 。 例如, 长度 1, 2, 3 无法设定三角形, 因为 1+2=3 , 因此它们都会位于同一行上。 长度 4, 5, 10 也不能设定三角形, 因为 4+5=9 < 10 。 请看下面的图片 :

  

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  The arcs show that the two sides would never meet to form a triangle.
  ::弧弧显示,双方决不会开会形成三角形。

  

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  To make a triangle, two sides must add up to be greater than the third side. This is called the Triangle Inequality Theorem. This means that if you know two sides of a triangle, there are only certain lengths that the third side could be. If two sides have lengths $\begin{array}{r}a\end{array}$ and $\begin{array}{r}b\end{array}$ , then the length of the third side, $\begin{array}{r}s\end{array}$ , has the range $\begin{array}{r}a-b<s<a+b.\end{array}$
  ::要形成三角形, 双边的加起来必须大于第三边。 这叫做三角不均论。 这意味着如果您知道三角形的两边, 则第三边可能只有一定的长度。 如果两边有 a 和 b 长度, 那么第三边的长度, s, 则有 a- b < < a+b > 的范围 。

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  What if you were given three lengths, like 5, 7 and 10? How could you determine if sides with these lengths form a triangle?
  ::如果给你们三个长度,比如5、7和10,你如何确定这些长度的两边是否构成三角形?

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  Examples
  ::实例

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  Example 1
  ::例1

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  Do the lengths 4.1, 3.5, 7.5 make a triangle? 
  ::长度 4.1, 3.5, 7.5 做三角形吗?

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  Use the Triangle Inequality Theorem. Check to make sure that the smaller two numbers add up to be greater than the largest number.
  ::使用三角不平等定理。 检查以确保小两个数字加起来大于最大数字 。

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  $\begin{array}{r}4.1+3.5>7.5\end{array}$  and  $\begin{array}{r}7.6>7.5\end{array}$  so y es these lengths make a triangle.
  ::4.1+3.5>7.5和7.6>7.5 所以这些长度是三角形。

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  Example 2
  ::例2

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  Do the lengths 4, 4, 8 make a triangle?
  ::4,4,8的长度是三角形吗?

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  Use the Triangle Inequality Theorem. Check to make sure that  the smaller two numbers add up to be greater than the largest number.
  ::使用三角不平等定理。 检查以确保小两个数字加起来大于最大数字 。

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  $\begin{array}{r}4+4=8\end{array}$ . No this is not a triangle because two lengths cannot equal the third.
  ::4+4=8. 不, 这不是三角形, 因为两个长度不能等于第三个长度 。

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  Example 3
  ::例3

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  Do the lengths 4, 11, 8 make a triangle?
  ::4、11、8的长度是三角形吗?

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  Use the Triangle Inequality Theorem. Check to make sure that the smaller two numbers add up to be greater than the largest number.
  ::使用三角不平等定理。 检查以确保小两个数字加起来大于最大数字 。

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  $\begin{array}{r}4+8=12\end{array}$ and $\begin{array}{r}12>11\end{array}$ so yes these lengths make a triangle.
  ::4+8=12 和 12>11, 所以这些长度是三角形 。

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  Example 4
  ::例4

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  Find the length of the third side of a triangle if the other two sides are 10 and 6.
  ::如果其他两边是10和6,则查找三角形第三侧的长度。

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  The Triangle Inequality Theorem can also help you find the range of the third side. The two given sides are 6 and 10. The third side, $\begin{array}{r}s\end{array}$ , must be between $\begin{array}{r}10-6=4\end{array}$ and $\begin{array}{r}10+6=16\end{array}$ . In other words, the range of values for $\begin{array}{r}s\end{array}$ is $\begin{array}{r}4<s<16\end{array}$ .
  ::三角不平等理论也可以帮助您找到第三侧的范围。 给定的两侧是 6 和 10 。 给定的第三侧, s, 必须在 10 - 6 = 4 和 10+ 6 = 16 之间。 换句话说, s 的值范围是 4 < s < 16 。

  

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  Notice the range is no less than 4, and not equal to 4. The third side could be 4.1 because $\begin{array}{r}4.1+6>10\end{array}$ . For the same reason, $\begin{array}{r}s\end{array}$ cannot be greater than 16, but it could 15.9, $\begin{array}{r}10+6>15.9\end{array}$ .
  ::通知范围不小于4,不等于4, 第三方可以是4.1, 因为4.1+6>10。 出于同样的原因, s不能大于16, 但它可能超过15.9, 10+6>15.9。

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  Example 5
  ::例5

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  The base of an isosceles triangle has length 24. What can you say about the length of each leg?
  ::等分三角形的底部有24长, 每条腿的长度你能说些什么?

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  To solve this problem, remember that an isosceles triangle has two congruent sides (the legs). We have to make sure that the sum of the lengths of the legs is greater than 24. In other words, if $\begin{array}{r}x\end{array}$ is the length of a leg:
  ::要解决这个问题, 请记住, 等分三角形有两个相容的两面( 腿 ) 。 我们必须确保腿长的总和大于 24 。 换句话说, 如果 x 是腿长 :

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  $$\begin{array}{rl}x+x& >24\\ 2x& >24\\ x& >12\end{array}$$

  
  ::x+x > 242x>24x>12

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  Each leg must have a length greater than 12.
  ::每条腿的长度必须大于12。

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  Review
  ::回顾

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  Determine if the sets of lengths below can make a triangle. If not, state why.
  ::确定下方的长度组能否建立三角形。 如果没有,请说明原因。

  1. 6, 6, 13
  2. 1, 2, 3
  3. 7, 8, 10
  4. 5, 4, 3
  5. 23, 56, 85
  6. 30, 40, 50
  7. 7, 8, 14
  8. 7, 8, 15
  9. 7, 8, 14.99

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  If two lengths of the sides of a triangle are given, determine the range of the length of the third side.
  ::如果给定三角形两边的两长,则决定第三边的长度范围。

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  10. 8 and 9
      ::8和9
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  11. 4 and 15
      ::4和15
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  12. 20 and 32
      ::20和32
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  13. 2 and 5
      ::2和5
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  14. 10 and 8
      ::第10和8条
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  15. $\begin{array}{r}x\end{array}$ and $\begin{array}{r}2x\end{array}$
      ::x 和 x 2x
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  16. The legs of an isosceles triangle have a length of 12 each. What can you say about the length of the base?
      ::等分形三角形的腿长度各为12。 对于基底的长度, 您能说什么 ?

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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 。

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  Resources
  ::资源