7.12 30-60-90三角

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  30-60-90三角

  As Lindsay was walking near the Civic Center she noticed a shadow on the ground. She looked at the shadow and then cast her eyes upward to see the top of a flagpole. “That flagpole is quite tall but I bet I had to look up a longer distance than the pole is tall,” Lindsay said to herself. “I would like to figure out the length of my line of sight from the ground to the top of the flagpole,” she added.

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  How can Lindsay use the diagram below to figure out the length of her line of sight?
  ::Lindsay怎么能用下面的图表 来弄清楚她视线的长度?

  

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  In this concept, you will learn about $\begin{align*}30^\circ-60^\circ-90^\circ\end{align*}$   right triangles .
  ::在这个概念中,你会学到 大约306090右三角。

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  Equilateral Triangle
  ::等边三角形

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  An is one that has three sides of equal length and three angles of equal measure . The angles of the triangle each measure 60°. When an is drawn in an equilateral triangle, it bisects the angle from which it is drawn and is a perpendicular bisector of the base. A perpendicular bisector of the base passes through the midpoint of the base and is perpendicular to that side. The first diagram shows an equilateral triangle and the second one shows an equilateral triangle and its altitude.
  ::是一个具有三面相等长度和三个角度相等的方形。三角形的角每个测量60度。当一个三角形被画成一个对等三角形时,它会将一个三角形所画的角分成两部分,并且是基座的直角两部分。一个基座的垂直两部分穿过基座的中点,与该边垂直。第一个图表显示一个对等三角形,第二个图表显示一个对等三角形及其高度。

  

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  In the second triangle the line $\begin{align*}\overline{CM}\end{align*}$  is the altitude of the equilateral triangle. The measure of $\begin{align*}\angle C\end{align*}$  has been divided into two angles each measuring 30°. The altitude meets the base of the triangle $\begin{align*}\overline{AB}\end{align*}$  at right angles and divides the base into $\begin{align*}\overline{AM}\end{align*}$  and $\begin{align*}\overline{MB}\end{align*}$  which are equal in length. If $\begin{align*}\overline{AM}+\overline{MB}=\overline{AB}\end{align*}$  then the length of $\begin{align*}\overline{AB}=x+x=2x\end{align*}$ . If $\begin{align*}\overline{AB}=\overline{AC}=\overline{BC}\end{align*}$  then $\begin{align*}\overline{AC}=2x\end{align*}$  and $\begin{align*}\overline{BC}=2x\end{align*}$ .
  ::在第二个三角形中,线的内线是等边三角形的高度。%C的度量已分成两个角度,每个角度的长度为30°。高度在右角度符合三角形的底部,并将基数分为长度相等的AM和MB。如果是AB=xxx,那么长度为AB=xx=2x。如果是AB=AC=BC,那么AC=2x和BC=2x。

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  The altitude of the equilateral triangle has divided it into two equal right triangles. $\begin{align*}\Delta BCM\end{align*}$  is a $\begin{align*}30^\circ-60^\circ-90^\circ\end{align*}$   triangle and it will be used to show the relationship between the legs and the hypotenuse of this special right triangle.
  ::等边三角形的高度已经将其分为两个等右三角形。 QBCM 是一个 30 60 90 三角形, 它将用来显示双腿与这个特殊右三角形的顺减关系 。

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  Let’s begin by using the Pythagorean Theorem to find the measure of the altitude $\begin{align*}\overline{CM}\end{align*}$ .
  ::让我们首先使用毕达哥里安神话来找到高度的度量。

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  First, write down the values of $\begin{align*}(a,b,c)\end{align*}$  for the Pythagorean Theorem.
  ::首先,写下毕达哥里安神话的(a,b,c)值。

  $$\begin{align*} \begin{array}{rcl} \overline{CM} &=& a \ = \ ? \\ \overline{MB} &=& b \ = \ x \\ \overline{BC} &=& c \ = \ 2x \end{array}\end{align*}$$

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  Next, write the Pythagorean Theorem and substitute in the values for $\begin{align*}(a,b,c)\end{align*}$ .
  ::接下来,写下《毕达哥里安神话论》,并替换(a,b,c)的值。

  $$\begin{align*}\begin{array}{rcl} c^2 &=& a^2+b^2 \\ (2x)^2 &=& a^2+(x)^2 \end{array}\end{align*}$$

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  Next, do the indicated squaring on both sides of the equation.
  ::接下来,在等式的两侧做标注的对角。

  $$\begin{align*}\begin{array}{rcl} (2x)^2 &=& a^2+(x)^2 \\ (2x \cdot 2x) &=& a^2+(x \cdot x) \\ 4x^2 &=& a^2+x^2 \end{array}\end{align*}$$

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  Next, isolate the variable by subtracting $\begin{align*}x^2\end{align*}$  from both sides of the equation and simplify.
  ::下一步,通过从方程两侧减去 x2 和简化来分离变量。

  $$\begin{align*} \begin{array}{rcl} 4x^2 &=& a^2+x^2 \\ 4x^2 - x^2 &=& a^2+x^2-x^2 \\ 3x^2 &=& a^2 \end{array}\end{align*}$$

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  Then, solve for the variable ‘ $\begin{align*}a\end{align*}$ ’ by taking the square root of both sides of the equation.
  ::然后,用方程式两侧的平方根解决变量“a”。

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  $$\begin{align*}\begin{array}{rcl} 3x^2 &=& a^2 \\ \sqrt{3} \cdot \sqrt{x^2} &=& \sqrt{a^2} \\ \sqrt{3} \cdot x &=& a \\ x \sqrt{3} &=& a \end{array}\end{align*}$$
  ::3x2 = a2322a23x=ax3=ax3=ax3=a

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  The answer is  $\begin{align*}x \sqrt{3}\end{align*}$ .
  ::答案是 x% 3 。

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  If  $\begin{align*}b=x\end{align*}$  then the answer can be expressed as  $\begin{align*}b \sqrt{3}\end{align*}$ .
  ::如果b=x,那么答案可以表示为b=3。

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  The relationship between the sides of the  $\begin{align*}30^\circ-60^\circ-90^\circ\end{align*}$  triangle is:
  ::306090三角形两侧的关系是:

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  • The length of the hypotenuse is twice the length of the shorter leg.
    ::下限长度是短腿长度的两倍。
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  • The length of the shorter leg is one-half the length of the hypotenuse.
    ::短腿的长度是下限长度的一半。
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  • The length of the longer leg is $\begin{align*}\sqrt{3}\end{align*}$  times the length of the shorter leg.
    ::长腿的长度是短腿的3倍。

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were given a problem about Lindsay and the flagpole shadow. She wants to know the length of her line of sight to the top of the flagpole.
  ::早些时候,你被问及林赛和旗杆阴影的问题。她想知道她的视线到旗杆顶部的长度。

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  Lindsay will have to use the relationships between the sides of a special triangle .
  ::林赛必须利用 特殊三角形两侧的关系

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  First, she should write down what she knows.
  ::首先,她应该写下她知道的

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  The length of the flagpole is the longer leg of the $\begin{align*}30^\circ-60^\circ-90^\circ\end{align*}$  triangle and is $\begin{align*}7 \sqrt{3} \ \text{feet}\end{align*}$  in length.
  ::旗杆的长度是306090三角形的长腿,长度为73英尺。

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  The length of the shadow is the shorter leg and its length is unknown.
  ::阴影的长度是短腿,其长度是未知的。

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  The length of the hypotenuse is the length of Lindsay’s line of sight and she wants to figure out this length.
  ::下限的长度是Lindsay的视线长度,她想弄清楚这一长度。

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  Next, identify the variables.
  ::接下来,确定变量。

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  Let ‘ $\begin{align*}b\end{align*}$ ’ represent the length of the longer leg, ‘ $\begin{align*}a\end{align*}$ ’ represent the length of the shorter leg and ‘ $\begin{align*}c\end{align*}$ ’ represent her line of sight.
  ::让 'b ' 代表长腿的长度, 'a ' 代表短腿的长度, 'c ' 代表她的视线。

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  Next, write down an equation showing the relationship between the two legs of the triangle.
  ::接下来,写下一个方程,显示三角形两腿之间的关系。

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  $$\begin{align*}b=\sqrt{3} \cdot a\end{align*}$$
  ::3a

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  Next, fill any known values into the equation.
  ::下一步,将任何已知值填入方程。

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  $$\begin{align*}\begin{array}{rcl} b &=& \sqrt{3} \cdot a \\ 7 \sqrt{3} &=& a \sqrt{3} \end{array}\end{align*}$$
  ::=====================================================================================================================================================================================================================================

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  Then, divide both sides of the equation by $\begin{align*}\sqrt{3}\end{align*}$  to solve for the variable ‘ $\begin{align*}a\end{align*}$ ’.
  ::然后,将方程的两边除以 3, 以解答变量“ a ” 。

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  $$\begin{align*}\begin{array}{rcl} 7 \sqrt{3} &=& a \sqrt{3} \\ \frac{7 \cancel{ \sqrt{3}}}{ \cancel{ \sqrt{3}}} &=& \frac{a \cancel{ \sqrt{3}}}{\cancel{ \sqrt{3}}}\\ 7 &=& a \end{array}\end{align*}$$
  ::7_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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  The answer is 7.
  ::答案是7个

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  The length of the shorter leg is 7 feet.
  ::短腿的长度是7英尺。

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  Next, write an equation showing the relationship between the hypotenuse and the shorter leg.
  ::接下来,写一个方程式,显示下限和短腿之间的关系。

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  $$\begin{align*}c=2a\end{align*}$$
  ::c=2 a

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  Next, substitute the length of the shorter leg into the equation.
  ::接下来,用短腿的长度来代替方程

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  $$\begin{align*}\begin{array}{rcl} c &=& 2a \\ c &=& 2(7) \end{array}\end{align*}$$
  ::c=2ac=2(7)

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  Then, perform the multiplication on the right side of the equation.
  ::然后在方程右侧进行乘法。

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  $$\begin{align*}\begin{array}{rcl} c &=& 2(7) \\ c &=& 14 \end{array}\end{align*}$$
  ::c=2(7)c=14

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  The answer is 14.
  ::答案是14岁

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  The length of Lindsay’s line of sight is 14 feet.
  ::Lindsay的视线长度为14英尺。

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  Example 2
  ::例2

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  For the following $\begin{align*}30^\circ-60^\circ-90^\circ\end{align*}$  triangle with a hypotenuse of 16 inches, find the length of the longer leg to the nearest hundredth.
  ::以下306090三角形的下限为16英寸的三角形,其长腿长到最近的一百英尺。

  

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  First, write down what you know from the diagram.
  ::首先,写下你从图表中知道的

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  The hypotenuse is  $\begin{align*}\overline{BC}=16 \ \text{inches}\end{align*}$
  ::下限为BC=16英寸

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  The shorter leg is $\begin{align*} \overline{AC}\end{align*}$ .
  ::短腿是AC。

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  The longer leg is $\begin{align*}\overline{AB}\end{align*}$ .
  ::长腿是AB。

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  Next, write an equation that shows the relationship between the shorter leg and the hypotenuse.
  ::接下来,写一个方程式, 显示短腿和下肢之间的关系。

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  $$\begin{align*}\overline{AC}= \frac{1}{2} (\overline{BC})\end{align*}$$
  ::AC=12(BC)

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  Then, substitute the value for $\begin{align*}\overline{BC}\end{align*}$  into the equation and simplify.
  ::然后,将BC的值换成方程,简化。

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  $$\begin{align*}\begin{array}{rcl} \overline{AC} &=& \frac{1}{2} (\overline{BC}) \\ \overline{AC} &=& \frac{1}{2} (16) \\ \overline{AC} &=& 8 \end{array}\end{align*}$$
  ::AC=12(BC)AC=12(16)AC=8

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  The answer is 8.
  ::答案是8岁

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  The length of the shorter leg is 8 inches.
  ::短腿长8英寸

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  Now determine the length of the longer leg $\begin{align*}\overline{AB}\end{align*}$ .
  ::现在确定长腿的长度

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  First, write an equation that shows the relationship between the two legs of the triangle.
  ::首先,写一个方程式,显示三角形两腿之间的关系。

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  $$\begin{align*}\overline{AB} = \overline{AC} \cdot \sqrt{3}\end{align*}$$
  
  Next, substitute the value of $\begin{align*}\overline{AC}\end{align*}$  into the equation and simplify.
  ::下一个,用方程式来取代AC的值 并简化

  $$\begin{align*}\begin{array}{rcl} \overline{AB} &=& \overline{AC} \cdot \sqrt{3} \\ \overline{AB} &=& (8) \cdot \sqrt{3} \\ \overline{AB} &=& 8 \sqrt{3} \end{array}\end{align*}$$

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  Then, use the calculator to express the length of the longer leg to the nearest hundredth.
  ::然后用计算器将长腿的长度 表示到最近的一百英尺

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  $$\begin{align*}\begin{array}{rcl} \overline{AB} &=& 8 \sqrt{3} \\ \overline{AB} &=& 8(1.73) \\ \overline{AB} &\approx& 13.8 \end{array}\end{align*}$$
  ::AB=8 3 AB=8 1 73 AB=13.8

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  The answer is approximately 13.8.
  ::答案约为13.8。

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  The length of the longer leg is approximately 13.8 inches.
  ::长腿长度约为13.8英寸。

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  Example 3
  ::例3

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  For the following $\begin{align*}30^\circ-60^\circ-90^\circ\end{align*}$  triangle with a longer leg of 6 inches, find the length of the shorter leg. Express the length as a radical .
  ::后面的306090 三角形的长腿6英寸, 找到短腿的长度。 将长度表示为激进 。

  

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  First, write down what you know from the diagram.
  ::首先,写下你从图表中知道的

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  The longer leg is  $\begin{align*}\overline{CD}=6 \ \text{inches}\end{align*}$
  ::长腿是6英寸

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  The shorter leg is $\begin{align*}\overline{EC}\end{align*}$ .
  ::短腿是EC。

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  The hypotenuse is $\begin{align*}\overline{ED}\end{align*}$ .
  ::下限为天线

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  Next, write an equation that shows the relationship between the longer leg and the shorter leg.
  ::接下来,写一个方程式,显示长腿和短腿之间的关系。

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  $$\begin{align*}\overline{CD}= \sqrt{3}(\overline{EC})\end{align*}$$
  ::CD3( EC)

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  Next, substitute the value for $\begin{align*}\overline{CD}\end{align*}$  into the equation and simplify.
  ::接下来,在方程中将 CD 的值替换为 CD , 并简化 。

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  $$\begin{align*} \begin{array}{rcl} \overline{CD} &=& \sqrt{3}(\overline{EC}) \\ 6 &=& \sqrt{3}(\overline{EC}) \end{array}\end{align*}$$
  ::===============================================================================================================================

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  Next, divide both sides of the equation by $\begin{align*}\sqrt{3}\end{align*}$  to solve  $\begin{align*}\overline{EC}\end{align*}$
  ::接下来,将方程的两侧除以 3 来解答 EC

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  $$\begin{align*}\begin{array}{rcl} 6 &=& \sqrt{3}(EC) \\ \frac{6}{\sqrt{3}} &=& \frac{ \cancel {\sqrt{ 3}} (EC)}{ \cancel {\sqrt{ 3}}} \\ \frac{6}{\sqrt{3}} &=& \overline{EC} \end{array}\end{align*}$$
  ::633(EC)63*3*3*3(EC)36*3=EC

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  Next, multiply the left side of the equation by $\begin{align*}\frac{\sqrt{3}}{\sqrt{3}}\end{align*}$  to clear the radical from the denominator.
  ::接下来,将方程左侧乘以 3 3 3 来清除分母中的激进。

  $$\begin{align*}\begin{array}{rcl} \frac{6}{ \sqrt{3}} &=& \overline{EC} \\ \frac{6}{ \sqrt{3}} \cdot \left( \frac{ \sqrt{3}}{ \sqrt{3}} \right ) &=& \overline{EC} \\ \frac{6 \sqrt{3}}{ \sqrt{9}}&=& \overline{EC} \end{array}\end{align*}$$  

  Next, simplify the left side of the equation.

  $$\begin{align*} \begin{array}{rcl} \frac{6 \sqrt{3}}{ \sqrt{9}}&=& \overline{EC} \\ \frac{6 \sqrt{3}}{ 3}&=& \overline{EC} \\ \frac{\overset{2}{\cancel{6} \sqrt{3}}}{\cancel{3}} &=& \overline{EC} \\ 2 \sqrt{3}&=& \overline{EC} \end{array} \end{align*}$$  

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  The answer is $\begin{align*}2 \sqrt{3}\end{align*}$ .
  ::答案是23

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  The length of the shorter leg is $\begin{align*}2 \sqrt{3}\end{align*}$  inches.
  ::短腿长2英寸3英寸

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  Example 4
  ::例4

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  For the following $\begin{align*}30^\circ-60^\circ-90^\circ\end{align*}$  triangle with a shorter leg of 4 feet, find the length of the hypotenuse to the nearest tenth.
  ::以下306090三角形的短腿4英尺,找到最接近10英尺的下限长度。

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  First, write down what you know from the diagram.
  ::首先,写下你从图表中知道的

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  The hypotenuse is  $\begin{align*}\overline{AB} \ = \ ? \ \text{feet}\end{align*}$
  ::双脚是双脚 双脚是双脚 双脚是双脚 双脚是双脚 双脚是双脚 双脚是双脚

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  The shorter leg is  $\begin{align*}\overline{AC} \ = \ 4 \ \text{feet}\end{align*}$
  ::短腿是AC=4英尺

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  The longer leg is $\begin{align*}\overline{BC}\end{align*}$ .
  ::长腿是BC。

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  Next, write an equation that shows the relationship between the shorter leg and the hypotenuse.
  ::接下来,写一个方程式, 显示短腿和下肢之间的关系。

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  $$\begin{align*}2(\overline{AC})=\overline{AB}\end{align*}$$
  :AC) 'A'

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  Then, substitute the value for $\begin{align*}\overline{AC}\end{align*}$  into the equation and simplify.
  ::然后,用方程式来取代 AC的值 并简化

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  $$\begin{align*}\begin{array}{rcl} 2(\overline{AC}) &=& \overline{AB} \\ 2(4) &=& \overline{AB} \\ 8 &=& \overline{AB} \end{array} \end{align*}$$
  :AC)=AB2(4)=AB8=AB

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  The answer is 8.
  ::答案是8岁

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  The length of the hypotenuse is 8 feet.
  ::低温的长度是8英尺

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  Review
  ::回顾

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  Find the missing length of the longer leg in each $\begin{align*}30^\circ-60^\circ - 90^\circ\end{align*}$  triangle.
  ::在每个306090三角形中查找缺失的长腿长度 。

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  1. Short leg = 3
  ::1. 短腿=3

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  2. Short leg = 4
  ::2. 短腿=4

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  3. Short leg = 2
  ::3. 短腿=2

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  4. Short leg = 8
  ::4. 短腿=8

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  5. Short leg = 10
  ::5. 短腿=10

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  Use a calculator to figure out the approximate value of each longer leg. You may round your answer when necessary.
  ::使用计算器来计算每条长腿的大约值。 必要时您可以绕过答案 。

  6. $\begin{align*}3 \sqrt{3}\end{align*}$

  7.  $\begin{align*}4 \sqrt{3}\end{align*}$

  8.  $\begin{align*}2 \sqrt{3}\end{align*}$

  9.  $\begin{align*}8 \sqrt{3}\end{align*}$

  10.  $\begin{align*}10 \sqrt{3} \end{align*}$

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  Use what you have learned to solve each problem.
  ::利用你学到的东西解决每个问题

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  11. Janie had construction paper cut into and equilateral triangle. She wants to cut it into two smaller congruent triangles. What will be the angle measurement of the triangles that result?
  ::11. Janie曾用建筑纸切入三角形和等边三角形,她想将其切成两个较小的相似三角形。

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  12. Madeleine has poster board in the shape of a square. She wants to cut two congruent triangles out of the poster board without leaving any leftovers. What will be the angle measurements of the triangles that result?
  ::12. 玛德琳的海报板形状是方形的,她想在不留下任何剩菜的情况下从海报板上切除两个相似的三角形。

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  13. A square window has a diagonal of $\begin{align*}2 \sqrt{3} \ \text{feet} \end{align*}$ . What is the length of the shorter of its legs?
  ::13. 方形窗口的对角为23英尺,短腿的长度是多少?

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  14. A square block of cheese is cut into two congruent wedges. If the shortest side of the original block was 9 inches, how long is the diagonal cut?
  ::14. 一块平方块的奶酪切成两块相似的网格,如果原块最短的一面是9英寸,对角的切口还要多久?

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  15. Jerry wants to find the area of a 30-60-90 triangle but only knows that the length of the shorter side is 4 centimeters. If the hypotenuse is 8 centimeters, what is the area of the triangle?
  ::15. Jerry想找到一个30-60-90三角的面积,但只知道较短的长度是4厘米,如果下限是8厘米,三角的面积是多少?

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