8.2 毕达哥里安神话和毕达哥里安三轮

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  毕达哥里安提奥伦和毕达哥里安三轮

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  Pythagorean Theorem
  ::毕达哥里安神论

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  The two shorter sides of a right triangle (the sides that form the right angle) are the legs and the longer side (the side opposite the right angle) is the hypotenuse . For the Pythagorean Theorem , the legs are “ $\begin{align*}a\end{align*}$ ” and “ $\begin{align*}b\end{align*}$ ” and the hypotenuse is “ $\begin{align*}c\end{align*}$ ”.
  ::右三角的两边(右角的两边)是双腿,右角对面的长边(右角对面的侧)是下限。 对俾达哥里安神话来说,双腿是“a”和“b”的,下限是“c”。

  

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  : Given a right triangle with legs of lengths $\begin{align*}a\end{align*}$ and $\begin{align*}b\end{align*}$ and a hypotenuse of length $\begin{align*}c\end{align*}$ , $\begin{align*}a^2+b^2=c^2\end{align*}$ .
  :::给定右三角形,长度为a和b,长度为c,a2+b2=c2。

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  The is also true. It allows you to prove that a triangle is a right triangle even if you do not know its angle measures.
  ::事实也是如此。 它允许您证明一个三角是一个右三角, 即使您不知道它的角度量 。

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  Pythagorean Theorem Converse: If the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.
  ::Pytagoren Theorem 反面:如果三角形最长的一面的正方形等于其他两面的正方形之和,则三角形是一个右三角形。

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  If $\begin{align*}a^2 + b^2 = c^2\end{align*}$ , then $\begin{align*}\triangle ABC\end{align*}$ is a right triangle.
  ::如果 a2+b2=c2, 则 ABC 是一个右三角形。

  

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  Pythagorean Triples
  ::毕达哥里亚三连

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  A combination of three numbers that makes the Pythagorean Theorem true is called a Pythagorean triple . Each set of numbers below is a Pythagorean triple.
  ::三个数字的组合使得毕达哥里安神话成为事实,称为毕达哥里安三联。下面的每组数字都是毕达哥里安三联。

  $$\begin{align*}3, 4, 5 && 5, 12, 13 && 7, 24, 25 && 8, 15, 17 && 9, 12, 15 && 10, 24, 26\end{align*}$$

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  Any multiple of a Pythagorean triple is also considered a Pythagorean triple. Multiplying 3, 4, 5 by 2 gives 6, 8, 10, which is another triple. To see if a set of numbers makes a Pythagorean triple, plug them into the Pythagorean Theorem.
  ::任何比达哥林三联体也被视为比达哥林三联体。乘以3, 4, 5乘以2等于6, 8, 10, 这是另外的三倍。 看看一组数字是否构成比达哥林三联体, 将其插入毕达哥林神话中。

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  What if you were told that a triangle had side lengths of 5, 12, and 13? How could you determine if the triangle were a right one?
  ::如果有人告诉你三角形的侧长为5、12和13,你如何确定三角形是否正确?

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  Examples
  ::实例

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  Example 1
  ::例1

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  What is the diagonal of a rectangle with sides 10 and 16?
  ::与第10和16边的矩形对角是什么?

  

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  For any square and rectangle, you can use the Pythagorean Theorem to find the length of a diagonal. Plug in the sides to find $\begin{align*}d\end{align*}$ .
  ::对于任何方形和矩形, 您可以使用 Pythagorean Theorem 来找到对角线的长度。 插在两侧以找到 d 。

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  $$\begin{align*}10^2 + 16^2 &= d^2\\ 100 + 256 &= d^2\\ 356 &= d^2\\ d &= \sqrt{356} = 2 \sqrt{89} \approx 18.87\end{align*}$$
  ::102+162=d2100+256=d2356=d2d356=28918.87

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  Example 2
  ::例2

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  Do 6, 7, and 8 make the sides of a right triangle?
  ::6、7和8是右三角的两边吗?

  

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  Plug the three numbers into the Pythagorean Theorem. Remember that the largest length will always be the hypotenuse, $\begin{align*}c\end{align*}$ . If $\begin{align*}6^2+7^2=8^2\end{align*}$ , then they are the sides of a right triangle.
  ::将三个数字插进 Pytagorean Theorem 中。 记住最大长度总是下限, c。 如果 62+72=82, 那么它们是右三角形的侧面 。

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  $$\begin{align*}6^2 + 7^2 &= 36 + 49=85\\ 8^2 &= 64 && 85 \neq 64, \ \text{so the lengths are not the sides of a right triangle.}\end{align*}$$
  ::62+72=36+49=8582=648564,所以长度不是右三角形的边。

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  Example 3
  ::例3

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  Find the length of the hypotenuse.
  ::查找下限长度 。

  

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  Use the Pythagorean Theorem. Set $\begin{align*}a = 8\end{align*}$ and $\begin{align*}b = 15\end{align*}$ . Solve for $\begin{align*}c\end{align*}$ .
  ::使用 Pytagorean 定理。 设置 a=8 和 b=15。 解决 c 。

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  $$\begin{align*}8^2 + 15^2 &= c^2\\ 64 + 225 &= c^2\\ 289 &= c^2 && Take \ the \ square \ root \ of \ both \ sides.\\ 17 &= c\end{align*}$$
  ::82+152=c264+225=c2289=c2 取两侧的平方根。

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  Example 4
  ::例4

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  Is 20, 21, 29 a Pythagorean triple?
  ::20 21 29是毕达哥伦三联赛吗?

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  If $\begin{align*}20^2 + 21^2 = 29^2\end{align*}$ , then the set is a Pythagorean triple.
  ::如果202+212=292,那么这套是毕达哥林三联赛。

  $$\begin{align*}20^2 + 21^2 &= 400+441=841\\ 29^2 &= 841\end{align*}$$

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  Therefore, 20, 21, and 29 is a Pythagorean triple.
  ::因此,20、21和29是毕达哥里三联赛

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  Example 5
  ::例5

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  Determine if the triangles below are right triangles.
  ::确定下方三角是否为右三角。

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  Check to see if the three lengths satisfy the Pythagorean Theorem. Let the longest side represent $\begin{align*}c\end{align*}$ .
  ::检查三个长度是否满足 Pytagorena 理论。 让最长的侧面代表 。

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  1. 
     $$\begin{align*}a^2 + b^2 & = c^2\\ 8^2 + 16^2 & \overset{?} = \left( 8 \sqrt{5} \right )^2\\ 64 + 256 & \overset{?} = 64 \cdot 5\\ 320 &= 320 \qquad \text{Yes}\end{align*}$$
     ::a2+b2=c282+162?=(85)264+256?=645320=320
  2. 

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  $$\begin{align*}a^2+b^2 & = c^2\\ 22^2 + 24^2 & \overset{?}= 26^2\\ 484 + 576 & \overset{?}= 676\\ 1060 & \neq 676 \qquad \text{No}\end{align*}$$
  ::a2+b2=c2222+242?=262484+576?=6761060=676

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  Review
  ::回顾

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  Find the length of the missing side. Simplify all radicals.
  ::查找缺失方的长度。 简化所有激进分子 。

  1. 
  2. 
  3. 
  4. 
  5. 
  6. 
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  7. If the legs of a right triangle are 10 and 24, then the hypotenuse is __________.
     ::如果右三角形的腿是10和24,那么下限是。
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  8. If the sides of a rectangle are 12 and 15, then the diagonal is _____________.
     ::如果矩形的两侧是12和15,那么对角就是。
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  9. If the sides of a square are 16, then the diagonal is ____________.
     ::如果方形的两边是16,那么对角就是...
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  10. If the sides of a square are 9, then the diagonal is _____________.
      ::如果方形两边是9,那么对角就是...

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  Determine if the following sets of numbers are Pythagorean Triples.
  ::确定以下一组数字是否是毕达哥伦三联赛。

  11. 12, 35, 37
  12. 9, 17, 18
  13. 10, 15, 21
  14. 11, 60, 61
  15. 15, 20, 25
  16. 18, 73, 75

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  Determine if the following lengths make a right triangle.
  ::确定以下长度是否为右三角形。

  17. 7, 24, 25
  18. $\begin{align*}\sqrt{5}, 2 \sqrt{10}, 3 \sqrt{5}\end{align*}$
  19. $\begin{align*}2 \sqrt{3}, \sqrt{6}, 8\end{align*}$
  20. 15, 20, 25
  21. 20, 25, 30
  22. $\begin{align*}8 \sqrt{3}, 6, 2 \sqrt{39}\end{align*}$

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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