Section: 使用毕达哥里定理的应用 | 11.10 使用毕达哥里定理的应用

Section outline

Applications Using the Pythagorean Theorem
::使用毕达哥里定理的应用

The Pythagorean Theorem and its converse have many applications for finding lengths and distances.
::毕达哥里安神话及其对立面有许多寻找长度和距离的申请。

Real-World Application: Diagonal Length
::真实世界应用程序: 对角长度

Maria has a rectangular cookie sheet that measures $\begin{align*}10 \ inches \times 14 \ inches\end{align*}$ . Find the length of the diagonal of the cookie sheet.
::Maria有一个长方形饼干页, 测量 10 英寸× 14 英寸。查找 cookie 页的对角长度。

Draw a sketch:
::绘制草图 :

Define variables: Let $\begin{align*}c =\end{align*}$ length of the diagonal.
::定义变量:让 c = 对角线的长度。

Write a formula : Use the Pythagorean Theorem: $\begin{align*}a^2+b^2=c^2\end{align*}$
::写入公式: 使用 Pydagorean 定理: a2+b2=c2

Solve the equation :
::解决方程式:

$\begin{aligned} 10^{2} + 14^{2} = c^{2} \\ 100 + 196 = c^{2} \\ c^{2} = 296 \Rightarrow c = \sqrt{296} \Rightarrow c = 2 \sqrt{74} or c = 17.2 i n c h e s \end{aligned}$
$\begin{align*}10^2+14^2 = c^2\\ 100+196 = c^2\\ c^2 = 296 \Rightarrow c=\sqrt{296} \Rightarrow c=2 \sqrt{74} \ \text{or} \ c = 17.2 \ inches\end{align*}$
::102+142=c2100+196=c2c2=296c=296c=296c=274或c=17.2英寸

Check: $\begin{align*}10^2+14^2=100+196=296\end{align*}$ and $\begin{align*}c^2=17.2^2=296\end{align*}$ . The solution checks out.
::检查:102+142=100+196=296和C2=17.22=296。

Real-World Application: Area of Shaded Regions
::实际世界应用:沙化区域面积

Find the area of the shaded region in the following diagram:
::在下图中查找阴影区域的面积:

Draw the diagonal of the square in the figure:
::绘制图中方形的对角线 :

Notice that the diagonal of the square is also the diameter of the circle.
::注意方形的对角线也是圆的直径。

Define variables: Let $\begin{align*}c =\end{align*}$ diameter of the circle.
::定义变量:让 C = 圆的直径。

Write the formula: Use the Pythagorean Theorem: $\begin{align*}a^2+b^2=c^2\end{align*}$ .
::写入公式: 使用 Pydagorean 理论: a2+b2=c2。

Solve the equation:
::解决方程式:

$\begin{aligned} 2^{2} + 2^{2} = c^{2} \\ 4 + 4 = c^{2} \\ c^{2} = 8 \Rightarrow c = \sqrt{8} \Rightarrow c = 2 \sqrt{2} \end{aligned}$
$\begin{align*}2^2+2^2 = c^2\\ 4+4 = c^2\\ c^2 = 8 \Rightarrow c=\sqrt{8} \Rightarrow c=2 \sqrt{2}\end{align*}$
::22+22=c24+4=c2c2=8*c=8*c=8*c=22

The diameter of the circle is $\begin{align*}2 \sqrt{2}\end{align*}$ , therefore the radius $\begin{align*}R=\sqrt{2}\end{align*}$ .
::圆的直径为22,因此半径为R=2。

Area of a circle formula: $\begin{align*}A=\pi \cdot R^2=\pi \left(\sqrt{2}\right)^2=2 \pi\end{align*}$ .
::圆形公式的面积: AR2(2)2=2。

The area of the shaded region is therefore $\begin{align*}2 \pi - 4 = 2.28\end{align*}$ .
::因此,阴影区域面积为24=2.28。

Real-World Application: Measuring the Sides of a Triangle
::真实世界应用:衡量三角边

In a right triangle, one leg is twice as long as the other and the perimeter is 28. What are the measures of the sides of the triangle?
::在右三角形中,一条腿是另一条腿的两倍长,周边是28号,三角形两侧的尺寸是多少?

Make a sketch and define variables:
::绘制草图和定义变量:

Let: $\begin{align*}a =\end{align*}$ length of the short leg
::Let: a=短腿长度

$\begin{align*}2a =\end{align*}$ length of the long leg
::2a=长腿长腿长度

$\begin{align*}c =\end{align*}$ length of the hypotenuse
::c=下限长度

Write formulas:
::写入公式 :

The sides of the triangle are related in two different ways.
::三角形的两侧有两种不同的关联

The perimeter is 28, so $\begin{align*}a+2a+c=28 \Rightarrow 3a+c=28\end{align*}$
::周边是28,所以一个+2a+c=28=28=3a+c=28

The triangle is a right triangle, so the measures of the sides must satisfy the Pythagorean Theorem:
::三角形是右三角形, 所以双方的测量必须满足毕达哥伦神话:

$\begin{aligned} a^{2} + (2 a)^{2} = c^{2} \Rightarrow a^{2} + 4 a^{2} = c^{2} \Rightarrow 5 a^{2} = c^{2} \\ or c = a \sqrt{5} = 2.236 a \end{aligned}$
$\begin{align*}&\qquad \qquad a^2+(2a)^2 = c^2 \Rightarrow a^2+4a^2=c^2 \Rightarrow 5a^2=c^2\\ &\text{or} \qquad \quad c = a\sqrt{5}=2.236a\end{align*}$
::a2+(2a)2=c2=c2=a2=a2+4a2=c2+5a2=c2=c2orc=a5=2.236a

Solve the equation:
::解决方程式:

Plug the value of $\begin{align*}c\end{align*}$ we just obtained into the perimeter equation: $\begin{align*}3a+c=28\end{align*}$
::插入我们刚刚获得的 c 值, 插入周界方程: 3a+c=28

$\begin{aligned} 3 a + 2.236 a = 28 \Rightarrow 5.236 a = 28 \Rightarrow a = 5.35 \end{aligned}$
$\begin{align*}3a+2.236a=28 \Rightarrow 5.236a=28 \Rightarrow a=5.35\end{align*}$
::3a+2.236a=285.236a=28a=5.35

The short leg is: $\begin{align*}a = 5.35\end{align*}$
::短腿是: a=5.35

The long leg is: $\begin{align*}2a = 10.70\end{align*}$
::长腿是: 2a=10.70

The hypotenuse is: $\begin{align*}c = 11.95\end{align*}$
::下限为: c=11.95

Check: The legs of the triangle should satisfy the Pythagorean Theorem:
::选中 : 三角形的腿应该满足 Pytagoren 理论 :

$\begin{align*}a^2+b^2=5.35^2+10.70^2=143.1, c^2=11.95^2=142.80.\end{align*}$ The results are approximately the same.
::a2+b2=53.52+10.702=143.1,c2=11.952=142.80。结果大致相同。

The perimeter of the triangle should be 28:
::三角形的周边应该是28:

$\begin{align*}a+b+c=5.35+10.70+11.95=28.\end{align*}$ The answer checks out.
::a+b+c=5.35+10.70+11.95=28。

Example
::示例示例示例示例

Example 1
::例1

Mike is loading a moving van by walking up a ramp. The ramp is 10 feet long and the bed of the van is 2.5 feet above the ground. How far does the ramp extend past the back of the van?
::麦克正在装上一辆移动的面包车,行驶在斜坡上。斜坡长10英尺,面包车的床在地面上2.5英尺。斜坡穿过面包车后方有多远?

Make a sketch:
::绘制一张草图:

Define variables: Let $\begin{align*}x =\end{align*}$ how far the ramp extends past the back of the van.
::定义变量:让 x = 坡道在面包车后侧延伸到多远。

Write a formula: Use the Pythagorean Theorem: $\begin{align*}x^2+2.5^2=10^2\end{align*}$
::写入公式: 使用 Pydagorean 定理: x2+2.52=102

Solve the equation:
::解决方程式:

$\begin{aligned} x^{2} + 6.25 & = 100 \\ x^{2} & = 93.5 \\ x & = \sqrt{93.5} = 9.7 f t \end{aligned}$
$\begin{align*}x^2+6.25 &= 100\\ x^2 &= 93.5\\ x &= \sqrt{93.5} = 9.7 \ ft\end{align*}$
::x2+6.25=100x2=93.5x=93.5=9.7英尺

Check by plugging the result in the Pythagorean Theorem:
::通过在 Pytagorean 理论中插入结果来检查结果 :

$\begin{align*}9.7^2+2.5^2=94.09+6.25=100.34 \approx 100\end{align*}$ . So the ramp is 10 feet long. The answer checks out.
::9.72+2.52=9.4.09+6.25=100.34100。所以斜坡长10英尺。答案检查出来。

Review
::回顾

In order to make a ramp that is $\begin{align*}3ft\end{align*}$ high and covers $\begin{align*}4ft\end{align*}$ of ground, how long must the ramp be?
::为了造一个高3英尺高的坡道,覆盖地面4英尺高的坡道,坡道需要多久?

A regulation baseball diamond is a square with 90 feet between bases. How far is second base from home plate?
::标准棒球钻石是距离基地90英尺的广场,离本垒盘还有多远?

Emanuel has a cardboard box that measures $\begin{align*}20 \ cm \ \text{long} \ \times \ 10 \ cm \ \text{wide} \ \times \ 8 \ cm \ \text{deep}\end{align*}$ .

What is the length of the diagonal across the bottom of the box?
::横跨盒子底部的对角线长度是多少?

What is the length of the diagonal from a bottom corner to the opposite top corner?
::从下角到对面顶角的对角长度是多少?

::伊曼纽尔有一个纸板盒,能测量20厘米长×10厘米宽×8厘米深。对角在盒子底部的长度是多少?对角从下角到对角的长度是多少?

Samuel places a ladder against his house. The base of the ladder is 6 feet from the house and the ladder is 10 feet long.

How high above the ground does the ladder touch the wall of the house?
::梯子在地上有多高触摸到房子的墙壁?

If the edge of the roof is 10 feet off the ground and sticks out 1.5 feet beyond the wall, how far is it from the edge of the roof to the top of the ladder?
::如果屋顶的边缘离地面10英尺远,在墙外停留1.5英尺,那么从屋顶边缘到梯子顶部还有多远?

::Samuel用梯子抵住他的房子。梯子的底部离房子6英尺,梯子10英尺长。梯子在地面上有多高?梯子触摸房子的墙壁?如果屋顶的边缘距地面10英尺远,而顶部的边缘距墙外1.5英尺,那么从屋顶的边缘到梯子的顶部还有多远呢?

Find the area of the triangle below if the area of a triangle is defined as $\begin{align*}A=\frac{1}{2} \ base \times height\end{align*}$ :
::如果三角形区域被定义为 A=12 基数x小时, 请查找下方三角形的区域 :

Instead of walking along the two sides of a rectangular field, Mario decided to cut across the diagonal. He thus saves a distance that is half of the long side of the field.

Find the length of the long side of the field given that the short side is 123 feet.
::找到野外长边的长度,因为短边是123英尺。

Find the length of the diagonal.
::查找对角线的长度。

::Mario没有沿着一个矩形场的两侧走动,而是决定跨越对角线。因此, 他节省了距离, 距离是地表长侧的一半。查找地表长侧的长度, 因为短边是123英尺。找到对角线的长度。

Marcus sails due north and Sandra sails due east from the same starting point. In two hours Marcus’ boat is 35 miles from the starting point and Sandra’s boat is 28 miles from the starting point.

How far are the boats from each other?
::船离彼此有多远?

Sandra then sails 21 miles due north while Marcus stays put. How far is Sandra from the original starting point?
::桑德拉离最初的起点还有多远?

How far is Sandra from Marcus now?
::桑德拉离马库斯有多远?

::马库斯向北航行,桑德拉从同一起点向东航行。两小时后,马库斯的船离起点35英里,桑德拉的船离起点28英里。船离起点有多远?桑德拉在马库斯停留时向北航行21英里。桑德拉离最初起点有多远?桑德拉现在离马库斯多远?

Determine the area of the circle below. (Hint: the hypotenuse of the triangle is the diameter of the circle.)
::确定以下圆的面积。 (提示:三角形的下限是圆的直径。 )

A rectangle's length is $\begin{align*}1in\end{align*}$ longer than its width and the diagonal has a length of $\begin{align*}29in\end{align*}$ . What are the lengths of the sides of the rectangle?
::矩形的长度比其宽度长1英寸,对角的长度为29英寸。矩形两侧的长度是多少?

For an isosceles triangle with sides of the length given, find the length of each hypotenuse:

1

2

3

$\begin{align*}n\end{align*}$
::n 内

::对于具有给定长度两边的等分三角形,可发现每个下限的长度:1 2 3 n

Review (Answers)
::回顾(答复)

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。

Section outline

Applications Using the Pythagorean Theorem ::使用毕达哥里定理的应用

Real-World Application: Diagonal Length ::真实世界应用程序: 对角长度

Real-World Application: Area of Shaded Regions ::实际世界应用:沙化区域面积

Real-World Application: Measuring the Sides of a Triangle ::真实世界应用:衡量三角边

Example ::示例示例示例示例

Example 1 ::例1

Review ::回顾

Review (Answers) ::回顾(答复)

Applications Using the Pythagorean Theorem
::使用毕达哥里定理的应用

Real-World Application: Diagonal Length
::真实世界应用程序: 对角长度

Real-World Application: Area of Shaded Regions
::实际世界应用:沙化区域面积

Real-World Application: Measuring the Sides of a Triangle
::真实世界应用:衡量三角边

Example
::示例示例示例示例

Example 1
::例1

Review
::回顾

Review (Answers)
::回顾(答复)