毕达哥里安神话的对立

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The Pythagorean Theorem and Energy
::毕达哥里理论和能源

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Mathematicians are employed in nearly every industry to use mathematics to find better and more efficient ways to complete tasks. This type of mathematician is called an applied mathematician. It’s surprising how mathematicians can apply so many math concepts to nearly any situation. Applied mathematicians find ways to use the Pythagorean theorem in situations that do not even involve right triangles . The following is an example of this.
::几乎所有行业都雇用数学家使用数学来寻找更好和更有效的完成任务的方法。 此类数学家被称为应用数学家。 令人惊讶的是数学家如何将如此众多的数学概念应用于几乎所有情况。 应用数学家在甚至不涉及右三角的情况下会找到使用Pythagorean定理的方法。 下面就是这方面的一个例子。

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What produces more power: one 10-volt battery or a 6-volt battery and an 8-volt battery combined?
::什么能产生更多动力:一个10伏电池或6伏电池和8伏电池结合?

 

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What is a Converse?
::什么是交锋?

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The converse is when you reverse the hypothesis and conclusion in a conditional statement . Look at this  example involving the statement “If I get an A, then I get ice cream.”
::反之亦然。 反之,当你在有条件的声明中推翻假设和结论时。 看看这个涉及“如果我得到A,那我就会得到冰淇淋”的声明的例子。

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This statement is an example of a converse that is not always true. If you get ice cream, it may be because you get an A, but it may have been because you wanted ice cream or any other number of reasons. In some cases the converse may be true in others it may be false.
::这个说法是一个反比的例子,它并不总是真实的。如果你得到了冰淇淋,那可能是因为你得到了A,但可能是因为你想要冰淇淋或其他原因。在某些情况下,反比在另一些情况下可能是真实的,它可能是虚假的。

 

 

 

 

 

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Prove It
::证明它

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Ancient Egyptians used a rope with evenly spaced knots to create right triangles. They knew that a triangle constructed with side lengths of 3, 4 and 5 knots would have to be a right triangle, so they used this rope to measure whether an angle was 90°. However, would this work for all side lengths of the relationship $\begin{align*}{a}^{2} + {b}^{2} = {c}^{2}?\end{align*}$
::古埃及人用一条绳子以平平的节节来创建右三角形。 他们知道一个侧长为3、4和5节的三角形必须是右三角形, 所以他们用这条绳子来测量角是否为90°。 但是, 这对关系 A2+b2=c2 的所有侧长都有效吗 ?

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This question refers to the converse of the Pythagorean Theorem. The original theorem states that “If a shape is a right triangle, then the sides must satisfy the relationship  $\begin{align*}a^2+b^2=c^2\end{align*}$  ”. The converse of this theorem states “If a the sides of a triangle satisfy the relationship   $\begin{align*}a^2+b^2=c^2\end{align*}$ , then it must be a right triangle”.
::这个问题指的是Pythagorean Theorem的反义词,原理论说,“如果一个形状是一个正确的三角形,那么两边必须满足A2+b2=c2的关系”。 该理论的反义说,“如果一个三角形的侧面满足了A2+b2=c2的关系,那么它必须是右三角形”。

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Use the interactive below to prove this relationship.
::使用下面的互动来证明这种关系。

 

 

 

 

 

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The Pythagorean Theorem and Energy Continued
::毕达哥里理论和能源继续

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Another way to look at the statement  $\begin{align*}{c}^{2} = {a}^{2} + {b}^{2}\end{align*}$  is that any square number can be written as the sum of two smaller squares numbers. For example, 25 can be written as 16 + 9, and 169 can be written as 25 + 144. The roots may not always be rational, but this relationship will always be possible. This theorem can be applied to nearly any situation involving two dimensions like area , force, and many more.
::另一种观察声明 c2=a2+b2 的方法是, 任何平方数字都可以以两个小平方数字的总和来写。 例如, 25可以写为 16 + 9, 169 可以写为 25 + 144 。 根可能并不总是合理, 但这种关系总是有可能的。 这个理论可以适用于任何涉及面积、 力和更多维等两个维度的情况 。

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One example of this involves . The amount of cardboard used to make one cube shaped box with a side length of 5 inches requires the same amount of cardboard as it would take to produce two cube-shaped boxes one with a side length of 4 and one with a side length of 3. You can check this by using the following formula: $\begin{align*}\text{surface area of a cube} = 6 \cdot \text{side length}^{2}.\end{align*}$
::其中一个例子涉及。用于制造一个侧长度为5英寸的立方体形状盒的纸板数量,需要与生产两个侧长度为4和侧长度为3的两个立方体形状盒所需的纸板数量相同。 您可以使用以下公式来检查:立方体表面面积=6度长度2。

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• The surface area of a 5-inch cubic box =  $\begin{align*}6 \cdot {5}^{2} = 150\end{align*}$ .
  ::5英寸立方箱的表面面积 = 652=150。
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• The surface area of a 4-inch cubic box =  $\begin{align*}6 \cdot {4}^{2} = 96\end{align*}$ .
  ::4英寸立方箱的表面面积 = 642=96。
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• The surface area of a 3-inch cubic box =  $\begin{align*}6 \cdot {3}^{2} = 54\end{align*}$ .
  ::3英寸立方箱的表面面积 = 632=54。

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Since $\begin{align*}54+96=150 \end{align*}$  and $\begin{align*}3^2+4^2=5^2 \end{align*}$ , you know that the surface area of the two smaller cubes is equal to the surface area of the larger cube.  This theorem can be applied to any two-dimensional value. In addition to identifying if a triangle contains a right angle , the Pythagorean theorem can also be used to determine if a triangle is acute or obtuse. Use the interactive below to see how.
::自54+96=150和32+42=52以来,你知道两个小立方体的表面积等于较大立方体的表面积。该定理可用于任何二维值。除了确定三角形是否包含右角度外,还可用Pytagorean定理来确定三角形是尖度还是模糊度。请使用下面的交互功能查看如何使用。

 

 

 

 

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Applied mathematicians can  apply  the Pythagorean Theorem to energy and batteries since power is a function of the square of voltage divided by resistance.
::应用数学家可以将毕达哥里安理论应用于能源和电池,因为电力是受抗力分割的电压平方的函数。

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Discussion Question : Using some of the strategies that you have learned thus far to determine what produces more power: one 10-volt battery or a 6-volt battery and an 8-volt battery combined?
::讨论问题:使用你迄今学到的一些战略来确定哪些发电量更多:一个10伏电池或一个6伏电池和一个8伏电池组合?

Summary

播放段落 A conditional statement is usually in the form of: If (hypothesis), then (conclusion). ::有条件声明通常采取下列形式:如果(假设),则(结论)。 播放段落 The converse of the statement is usually in the form of: If (conclusion), then (hypothesis). ::声明的相反形式通常为:如果(结束),那么(假设)。