1.7 特殊三角比率

Section outline

• Select section General

  Collapse Expand

  General

  Collapse all Expand all

  • Select activity 新闻通告

    

    新闻通告 Forum
• Select section 特殊三角比率

  Collapse Expand

  特殊三角比率

  播放段落

  While working in your Industrial Arts class one day, your Instructor asks you to use your 45-45-90 triangle to make a scale drawing. Unfortunately, you have two differently shaped triangles to use at your drafting table, and there aren't labels to tell you which triangle is the correct one to use.
  ::您的教官一天在工业艺术课上工作时, 您的教官要求您使用您的 45 - 45 - 90 三角来绘制比例尺。 不幸的是, 您在起草表格上需要使用两个不同形状的三角形, 而没有标签可以告诉你哪个三角形是正确的三角形 。

  播放段落

  You turn the triangles over and over in your hands, trying to figure out what to do, when you spot the ruler at your desk. Taking one of the triangles, you measure two of its sides. You determine that the first side is 7 inches long, and the second side is just a little under 9.9 inches. Can you determine if this is the correct triangle for your work?
  ::将三角形翻转在你的手中, 试图找出要做什么, 当你在桌子上看到标尺时。 取一个三角形, 你测量其两边。 你确定第一边是7英寸长, 第二边是略低于9. 9英寸。 你能确定这是你工作的正确三角形吗 ?

  播放段落

  Special Triangle Ratios
  ::特殊三角比率

  播放段落

  Special right triangles are the basis of trigonometry . The angles $\begin{array}{r}{30}^{\circ },{45}^{\circ },{60}^{\circ }\end{array}$ and their multiples have special properties and significance in the unit circle (which you can read about in other Concepts). Students are usually required to memorize the ratios of sides in triangles with these angles because of their importance.
  ::特殊右三角是三角测量的基础。 角度 30 、 45 、 60 及其多重在单位圆中具有特殊属性和意义( 您可以在其他概念中读到 ) 。 学生通常需要用这些角度对三角形的边边比进行记忆, 因为它们很重要 。

  播放段落

   First, let’s compare the two ratios, so that we can better distinguish the difference between the two. For a $\begin{array}{r}45-45-90\end{array}$ triangle the ratio is $\begin{array}{r}x:x:x\sqrt{2}\end{array}$ and for a $\begin{array}{r}30-60-90\end{array}$ triangle the ratio is $\begin{array}{r}x:x\sqrt{3}:2x\end{array}$ . An easy way to tell the difference between these two ratios is the isosceles right triangle has two congruent sides, so its ratio has the $\begin{array}{r}\sqrt{2}\end{array}$ , whereas the $\begin{array}{r}30-60-90\end{array}$ angles are all divisible by 3, so that ratio includes the $\begin{array}{r}\sqrt{3}\end{array}$ . Also, if you are ever in doubt or forget the ratios, you can always use the Pythagorean Theorem . The ratios are considered a short cut.
  ::首先,让我们比较这两个比率,这样我们就可以更好地区分两者之间的差异。对于45-45-90三角形,这个比率是x:x:x2,对于30-60-90三角形,这个比率是x:x3:2x。一个简单的方法可以分辨这两个比率之间的差别,就是等分层右边的三角形有两个相近的两面,因此其比率是2,而30-60-90角角是3分的,因此,这个比率包括3,如果你怀疑或忘记比率,你也可以使用Pythagoren Theorem。这些比率被认为是短路。

  播放段落

  Identifying Special Triangles 
  ::确定特殊三角

  播放段落

  Determine if the set of lengths represents a special right triangle. If so, which one?
  ::确定一组长度是否代表一个特殊的右三角形。 如果是, 哪一个?

  1.  $\begin{array}{r}8\sqrt{3}:24:16\sqrt{3}\end{array}$

  播放段落

  Yes, this is a $\begin{array}{r}30-60-90\end{array}$ triangle. If the short leg is $\begin{array}{r}x=8\sqrt{3}\end{array}$ , then the long leg is $\begin{array}{r}8\sqrt{3}\cdot \sqrt{3}=8\cdot 3=24\end{array}$ and the hypotenuse is $\begin{array}{r}2\cdot 8\sqrt{3}=16\sqrt{3}\end{array}$ .
  ::是的, 这是一个 30- 60- 90 三角形。 如果短腿为 x= 83, 那么长腿为 83- 3= 8- 3= 24, 下限为 2Q- 83= 163 。

  2.  $\begin{array}{r}\sqrt{5}:\sqrt{5}:\sqrt{10}\end{array}$

  播放段落

   Yes, this is a $\begin{array}{r}45-45-90\end{array}$ triangle. The two legs are equal and $\begin{array}{r}\sqrt{5}\cdot \sqrt{2}=\sqrt{10}\end{array}$ , which would be the length of the hypotenuse.
  ::是的, 这是 45- 45- 90 三角形, 双腿相等, 52= 10, 也就是下限长度 。

  3.  $\begin{array}{r}6\sqrt{7}:6\sqrt{21}:12\end{array}$

  播放段落

  No, this is not a special right triangle. The hypotenuse should be $\begin{array}{r}12\sqrt{7}\end{array}$ in order to be a $\begin{array}{r}30-60-90\end{array}$ triangle.
  ::不,这不是一个特殊的右三角形。 下限值应该是 127 , 才能成为 30- 60- 90 三角形 。

  播放段落

  Examples
  ::实例

  播放段落

  Example 1
  ::例1

  播放段落

  Earlier, you were asked if you have the right triangle for your work.
  ::早些时候,有人问您是否拥有工作所需的合适的三角形。

  播放段落

  Since you know the ratios of lengths of sides for special triangles , you can test to see if the triangle in your hand is the correct one by testing the relationship:
  ::由于您知道特殊三角形边边的长度比例, 您可以测试您手中的三角形是否正确, 测试此关系 :

  播放段落

  hypotenuse = $\begin{array}{r}\sqrt{2}x\end{array}$
  ::光电 = 2x

  播放段落

  where "x" is the length of the shorter sides. If you test this relationship with the triangle you are holding:
  ::这里的“ x” 是短边的长度。 如果您测试此与三角形的关系, 您将持有 :

  播放段落

  hypotenuse = $\begin{array}{r}7\sqrt{2}=9.87\end{array}$ in
  ::= 72=9.87英寸

  播放段落

  Yes, you are holding the correct triangle.
  ::是的,你拿着正确的三角形

  播放段落

  Example 2
  ::例2

  播放段落

   Determine if the set of lengths below represents a special right triangle. If so, which one?
  ::确定下方的长度组是否代表一个特殊的右三角形。 如果是,是哪一个?

  $\begin{array}{r}3\sqrt{2}:3\sqrt{2}:6\end{array}$

  播放段落

   The sides are the same length. This means that if the triangle is one of the special triangles at all, it must be a 45-45-90 triangle. To test this, we take either of the sides that are equal and multiply it by $\begin{array}{r}\sqrt{2}\end{array}$ :
  ::两边长度相同。 这意味着, 如果三角形是特殊三角形之一, 则必须是一个 45- 45- 90 三角形。 要测试这一点, 我们选取相等的两边, 乘以 2 :

  $\begin{array}{r}3\sqrt{2}\times \sqrt{2}=3\times \sqrt{4}=3\times 2=6\end{array}$

  播放段落

  Yes, this triangle is a special triangle . It is a 45-45-90 triangle.
  ::是的,三角形是一个特殊的三角形。这是一个45 -45 -90三角形。

  播放段落

  Example 3
  ::例3

  播放段落

  Determine if the set of lengths below represents a special right triangle. If so, which one?
  ::确定下方的长度组是否代表一个特殊的右三角形。 如果是,是哪一个?

  $\begin{array}{r}4:2:2\sqrt{3}\end{array}$

  播放段落

   It can immediately be seen that the second side is one half the length of the first side. This means that if it is a special triangle, it must be a 30-60-90 triangle . To see if it is indeed such a triangle, look at the relationship between the shorter side and the final side. The final side is $\begin{array}{r}\sqrt{3}\end{array}$ times the short side. So yes, this fulfills the criteria for a 30-60-90 triangle.
  ::可以看到第二侧是第一侧长度的一半。 这意味着如果它是特殊的三角形, 它必须是30- 60- 90三角形。 要看它是否真的是一个三角形, 请看看短侧和最后一面之间的关系。 最后一面是短侧的3倍。 所以是的, 这符合 30- 60- 90 三角形的标准 。

  播放段落

  Example 4
  ::例4

  播放段落

  Determine if the set of lengths below represents a special right triangle. If so, which one?
  ::确定下方的长度组是否代表一个特殊的右三角形。 如果是,是哪一个?

  $\begin{array}{r}13:84:85\end{array}$

  播放段落

  It can be seen immediately that the lengths of sides given aren't a special triangle, since 84 is so close to 85. Therefore it can't be a 45-45-90 triangle, which would require $\begin{array}{r}84\sqrt{2}\end{array}$ to be a side or a 30-60-90 triangle, where a one of these two sides would have a relationship of multiplying/dividing by 2 or by $\begin{array}{r}\sqrt{3}\end{array}$ .
  ::可以立即看出,双方的长度不是一个特殊的三角形, 因为84是接近85的, 因此它不可能是一个45 - 45 - 90三角形, 需要842是侧方或30 - 60 - 90三角形, 其中一方会以2或3成乘/分为关系。

  播放段落

  Review
  ::回顾

  播放段落

  For each of the set of lengths below, determine whether or not they represent a special right triangle. If so, which one?
  ::对于以下每一组长度,确定它们是否代表一个特殊的右三角。 如果是,哪一个?

  1. $\begin{array}{r}2:2:2\sqrt{2}\end{array}$
  2. $\begin{array}{r}3:3:6\end{array}$
  3. $\begin{array}{r}3:3\sqrt{3}:6\end{array}$
  4. $\begin{array}{r}4\sqrt{2}:4\sqrt{2}:8\sqrt{2}\end{array}$
  5. $\begin{array}{r}5\sqrt{2}:5\sqrt{2}:10\end{array}$
  6. $\begin{array}{r}7:7\sqrt{2}:14\end{array}$
  7. $\begin{array}{r}6\sqrt{5}:18\sqrt{5}:12\sqrt{5}\end{array}$
  8. $\begin{array}{r}4\sqrt{6}:12\sqrt{2}:8\sqrt{6}\end{array}$
  9. $\begin{array}{r}8\sqrt{15}:24\sqrt{5}:16\end{array}$
  10. $\begin{array}{r}7\sqrt{6}:7\sqrt{6}:14\sqrt{3}\end{array}$
  11. $\begin{array}{r}5\sqrt{7}:5\sqrt{14}:5\sqrt{7}\end{array}$
  12. $\begin{array}{r}9\sqrt{6}:27\sqrt{2}:18\sqrt{6}\end{array}$

  播放段落
  13. Explain why if you cut any square in half along its diagonal you will create two 45-45-90 triangles.
      ::解释一下,如果你把方块切成两半 沿着对角,你为何要建立两个45 -45 -90三角形?
  播放段落
  14. Explain how to create two 30-60-90 triangles from an equilateral triangle.
      ::解释如何从等边三角形创建两个 30 -60 -90 三角形 。
  播放段落
  15. Could a special right triangle ever have all three sides with integer lengths?
      ::一个特殊的右三角 是否有三面的整形长度?

  播放段落

  Review (Answers)
  ::回顾(答复)

  播放段落

  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 。