Section: 使用毕达哥里定理 | 7.8 使用毕达哥里定理

Section outline

The owners of a campground have just purchased a small island that they hope to add to their property as a spot for campers who use tents. First, the owners need to build a bridge to connect the two properties. The river is 34 feet wide and the higher land on the island is 13 feet from the river. How can the owners figure out the length of the bridge they need to build?
::营地地主刚刚购买了一座小岛,他们希望将小岛加到他们的地产上,作为使用帐篷的露营者的一个地点。首先,房主需要建一座桥梁,连接这两个地产。河宽34英尺,岛上高地距离河13英尺。房主如何知道他们需要修建的桥梁的长度?

In this concept, you will learn to use the Pythagorean Theorem .
::在这个概念中,你会学会使用毕达哥里安神话。

Indirect Measurement
::间接计量

In the real world, it is often not possible to use determine distances or lengths using rulers, tape measures or other measuring devices. When direct measurement is not an option, is used. Indirect measurement is an approach to measuring using alternative measurements and properties that will assist in determining the unknown distances or lengths needed to be found. The properties of the Pythagorean Theorem are often used when something needs to be indirectly measured.
::在现实世界中,通常无法使用标尺、磁带测量或其他测量装置来决定距离或长度。当直接测量不是一个选项时,则使用间接测量。间接测量是一种使用替代测量和属性来测量的方法,有助于确定需要找到的未知距离或长度。当需要间接测量某物时,往往使用Pythagoren定理仪的特性。

Consider the following situation.
::审议以下情况。

Jim must clean the rain gutters of his house. The gutters are 12 feet above the ground. He knows the ladder must extend 3 feet above the gutters so he can hold unto the rails as he cleans. If he places the foot of the ladder 4 feet from the base of the house, what length should the ladder be that he uses?
::吉姆必须清理他家的雨水槽。沟槽在地上12英尺。他知道梯子必须在沟槽上方3英尺宽,这样他就可以在打扫时守在铁轨上。如果他把梯子的脚从房子底部的4英尺移开,那么梯子应该用多久?

First, draw a diagram to represent the information.
::首先,绘制显示信息的图表。

Next, determine the values for $\begin{aligned} (a, b, c) \end{aligned}$ of the Pythagorean Theorem.
::下一步,确定毕达哥里安理论(a、b、c)的值。

$\begin{aligned} \begin{array}{rcl} a & = & 12 feet + 3 feet = 15 \\ b & = & 4 feet = 4 \\ c & = & length of ladder \end{array} \end{aligned}$

::a=12英尺+3英尺=15b=4英尺=4c=梯子长度

Next, fill the values into the Pythagorean Theorem.
::接下来,将值填入毕达哥里安理论。

$\begin{aligned} \begin{array}{rcl} c^{2} & = & a^{2} + b^{2} \\ c^{2} & = & (15)^{2} + (4)^{2} \end{array} \end{aligned}$

Next, perform the indicated squaring and simplify the equation.
::c2=a2+b2c2=(152+(4)2)

$\begin{aligned} \begin{array}{rcl} c^{2} & = & (15)^{2} + (4)^{2} \\ c^{2} & = & (15 \times 15) + (4 \times 4) \\ c^{2} & = & 225 + 16 \\ c^{2} & = & 241 \end{array} \end{aligned}$

Then, solve for $\begin{aligned} c \end{aligned}$ by taking the square root of both sides of the equation.
::c2=(1552)+(4)2c2=(15x15)+(4x4)c2=225+16c2=241 然后,用方程两侧的平方根解决 c。

$\begin{aligned} \begin{array}{rcl} c^{2} & = & 241 \\ \sqrt{c^{2}} & = & \sqrt{241} \\ c & = & 15.52 \end{array} \end{aligned}$

::c2=241c2=241c=15.52

The answer is 15.52.
::答案是15.52。

Jim needs to use a 16 foot ladder.
::吉姆需要用16英尺的梯子

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about the campgrounds and the island. The owners have to determine the length of the bridge they need to build to connect the campgrounds to the island. How can they calculate this indirect measurement using the Pythagorean Theorem?
::早些时候,有人给了你一个关于露营地和岛屿的问题。所有者必须确定他们为连接露营地和岛屿而需要修建的桥梁的长度。他们如何用毕达哥伦神话来计算间接测量?

First, draw and label a right triangle to model the problem.
::首先,绘制并标出一个右三角形来模拟问题。

Next, determine the values of $\begin{aligned} (a, b, c) \end{aligned}$ for the Pythagorean Theorem.
::下一步,为毕达哥里安神话确定(a、b、c)值。

$\begin{aligned} \begin{array}{rcl} a & = & 13 f t = 13 \\ b & = & 34 f t = 34 \\ c & = & ? f t = c \end{array} \end{aligned}$

Next, fill the values into the Pythagorean Theorem.
::a=13英尺 = 13b= 34英尺 = 34c=?ft= c 下一个,将值填入毕达哥里安定理词中。

$\begin{aligned} \begin{array}{rcl} c^{2} & = & a^{2} + b^{2} \\ c^{2} & = & (13)^{2} + (34)^{2} \end{array} \end{aligned}$

Next, perform the indicated squaring and simplify the equation.
::c2=a2+b2c2=(13)2+(34)2

$\begin{aligned} \begin{array}{rcl} c^{2} & = & 13^{2} + (34)^{2} \\ c^{2} & = & (13 \times 13) + (34 \times 34) \\ c^{2} & = & 169 + 1156 \\ c^{2} & = & 1325 \end{array} \end{aligned}$

Then, solve for $\begin{aligned} c \end{aligned}$ by taking the square root of both sides of the equation.
::c2=132+(342)2c2=(13x13)13+(34x34)c2=169+1156c2=1325 然后,用方程两侧的平方根解决 c。

$\begin{aligned} \begin{array}{rcl} c^{2} & = & 1325 \\ \sqrt{c^{2}} & = & \sqrt{1325} \\ c & = & 36.4 \end{array} \end{aligned}$

::c2=1325c2=1325c=36.4

The answer is 36.4.
::答案是36.4

The length of the bridge must be 36.4 feet.
::桥的长度必须是36.4英尺。

Example 2
::例2

The new Sunset Walking Park has just opened and you would like to take your mom to see it. The park is in the shape of a rectangle measuring 2 miles by 3 miles. You don’t know whether to take your mom for a walk around the outside perimeter of the park or to take her along the diagonal pathway. Which way would be shorter and by how much?
::新的日落漫步公园刚刚开放,你愿意带你妈妈去看它。公园的形状是两英里乘三英里的矩形。你不知道是带你妈妈在公园外围散步,还是带她走对角路。哪条路更短,多少路?

First, draw and label a diagram to represent the park.
::首先,绘制并标注一张代表公园的图表。

You can see that the diagonal pathway divides the rectangle into two right triangles. Use to find the length of the diagonal pathway.
::您可以看到对角路径将矩形分割为两个右三角。用来查找对角路径的长度。

Next, determine the values of $\begin{aligned} (a, b, c) \end{aligned}$ for the Pythagorean Theorem.
::下一步,为毕达哥里安神话确定(a、b、c)值。

$\begin{aligned} \begin{array}{rcl} a & = & 3 miles = 3 \\ b & = & 2 miles = 2 \\ c & = & ? miles = c \end{array} \end{aligned}$

Next, fill the values into the Pythagorean Theorem.
::a=3英里 = 3b=2英里 = 2c=? 英里 = c 下一步,将数值填入毕达哥里安理论。

$\begin{aligned} \begin{array}{rcl} c^{2} & = & a^{2} + b^{2} \\ c^{2} & = & (3)^{2} + (2)^{2} \end{array} \end{aligned}$

::c2=a2+b2c2=(3)2+(2)2

Next, perform the indicated squaring and simplify the equation.
::下一步,执行指示的对角并简化方程。

$\begin{aligned} \begin{array}{rcl} c^{2} & = & (3)^{2} + (2)^{2} \\ c^{2} & = & (3 \times 3) + (2 \times 2) \\ c^{2} & = & 9 + 4 \\ c^{2} & = & 13 \end{array} \end{aligned}$

::c2=(3)2+(2)2c2=(3x3)+(2x2)c2=9+4c2=13

Then, solve for $\begin{aligned} c \end{aligned}$ 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C问题

$\begin{aligned} \begin{array}{rcl} c^{2} & = & 13 \\ \sqrt{c^{2}} & = & \sqrt{13} \\ c & = & 3.6 \end{array} \end{aligned}$

::c2=13c2=13c=3.6

The answer is 3.6.
::答案是3.6。

The length of the pathway is 3.6 miles.
::这条通道的长度是3.6英里

The distance around the park is the sum of the lengths of the four sides of the rectangle.
::公园周围的距离是矩形四侧长度的总和。

$\begin{aligned} d = 3 + 2 + 3 + 2 \\ d = 10 \end{aligned}$

::d=3+2+3+2d=10 d=3+2+3+2d=10

The answer is 10.
::答案是10个

The distance around the park is 10 miles.
::公园周围的距离是10英里

$\begin{aligned} 10.0 miles - 3.6 miles = 6.4 miles \end{aligned}$

::10.0英里-3.6英里=6.4英里

The answer is 6.4 miles.
::答案是6.4英里

Example 3
::例3

An isosceles triangle has sides 30 inches, 30 inches and 20 inches. Find the length to the shortest side of the triangle using Pythagorean Theorem. Express the length of the altitude to the nearest inch.
::等星三角形的侧面为 30 英寸、 30 英寸和 20 英寸。使用 Pytagoren 理论来查找三角形最短的侧面的长度。将高度长度表示为最近的英寸。

First, draw and label a diagram to represent the problem.
::首先,绘制并标注一个图表来代表问题。

The altitude of an isosceles triangle cuts the base of the triangle into two equal lengths and creates two right triangles.
::等分三角形的高度将三角形的底部切成两个相等的长度,并创建两个右三角形。

Next, determine the values of $\begin{aligned} (a, b, c) \end{aligned}$ for the Pythagorean Theorem.
::下一步,为毕达哥里安神话确定(a、b、c)值。

$\begin{aligned} \begin{array}{rcl} a & = & 10 in = 10 \\ b & = & ? in = b \\ c & = & 30 in = 30 \end{array} \end{aligned}$

Next, fill the values into the Pythagorean Theorem.
::接下来,将值填入毕达哥里安理论。

$\begin{aligned} \begin{array}{rcl} c^{2} & = & a^{2} + b^{2} \\ (30)^{2} & = & (10)^{2} + b^{2} \end{array} \end{aligned}$

Next, perform the indicated squaring and simplify the equation.
::下一步,执行指示的对角并简化方程。

$\begin{aligned} \begin{array}{rcl} (30)^{2} & = & (10)^{2} + b^{2} \\ (30 \times 30) & = & (10 \times 10) + b^{2} \\ 900 & = & 100 + b^{2} \end{array} \end{aligned}$

Next, isolate the variable by subtracting 100 from both sides of the equation and simplify the equation.
::其次,从方程的两侧减去100,将变量分离出来,简化方程。

$\begin{aligned} \begin{array}{rcl} 900 & = & 100 + b^{2} \\ 900 - 100 & = & 100 - 100 + b^{2} \\ 800 & = & b^{2} \end{array} \end{aligned}$

Then, solve for the variable by taking the square root of both sides of the equation.
::然后,用方程式两侧的平方根来解决变量。

$\begin{aligned} \begin{array}{rcl} 800 & = & b^{2} \\ \sqrt{800} & = & \sqrt{b^{2}} \\ 28.28 & = & b \end{array} \end{aligned}$

The answer is 28.
::答案是28岁

The length of the altitude is 28 inches.
::高度为28英寸

Example 4
::例4

Find the length of the hypotenuse of a right triangle when the lengths of the other two sides are 5 m and 12 m.
::当其他两边的长度为5米和12米时,查找右三角形的下方长度。

First, draw and label a right triangle to model the problem.

::首先,绘制并标出一个右三角形来模拟问题。

Next, determine the values of $\begin{aligned} (a, b, c) \end{aligned}$ for the Pythagorean Theorem.
::下一步,为毕达哥里安神话确定(a、b、c)值。

$\begin{aligned} \begin{array}{rcl} a & = & 5 m = 5 \\ b & = & 12 m = 12 \\ c & = & ? m = c \end{array} \end{aligned}$

Next, fill the values into the Pythagorean Theorem.
::接下来,将值填入毕达哥里安理论。

$\begin{aligned} \begin{array}{rcl} c^{2} & = & a^{2} + b^{2} \\ c^{2} & = & (5)^{2} + (12)^{2} \end{array} \end{aligned}$

Next, perform the indicated squaring and simplify the equation.
::下一步,执行指示的对角并简化方程。

$\begin{aligned} \begin{array}{rcl} c^{2} & = & (5)^{2} + (12)^{2} \\ c^{2} & = & (5 \times 5) + (12 \times 12) \\ c^{2} & = & 25 + 144 \\ c^{2} & = & 169 \end{array} \end{aligned}$

Then, solve for $\begin{aligned} c \end{aligned}$ 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C问题

$\begin{aligned} \begin{array}{rcl} c^{2} & = & 169 \\ \sqrt{c^{2}} & = & \sqrt{169} \\ c & = & 13 \end{array} \end{aligned}$

The answer is 13.
::答案是13岁

The length of the hypotenuse is 13 m.
::下限长度为13米。

Review
::回顾

Use the Pythagorean Theorem to determine if each set of measurements could represent a right triangle. Sets of integers that could represent the sides of right triangles are called Pythagorean Triples.
::使用 Pytagorean 理论来决定每组测量是否代表右三角。能够代表右三角形两侧的整数组称为 Pytagoren Triples 。

1. 3, 4, 5

2. 6, 8, 12

3. 6, 8, 10

4. 15, 20, 25

5. 5, 9, 14

6. 9, 12, 15

7. 18, 24, 30

8. 5, 12, 13

9. 7, 24, 25

10. 8, 15, 17

Find the missing side length of each right triangle by using the Pythagorean Theorem. You may round to the nearest tenth when necessary.
::使用 Pytagorena 定理来查找右三角形的侧边长度。必要时您可以绕到最近的 10 。

11. $\begin{aligned} a = 6, b = 10, c = \underline{} \end{aligned}$
::11. a=6,b=10,c

12. $\begin{aligned} a = 5, b = 7, c = \underline{} \end{aligned}$
::12. a=5,b=7,c

13. $\begin{aligned} a = 7, b = 9, c = \underline{} \end{aligned}$
::13. a=7,b=9,c

14. $\begin{aligned} a = 6, b = 8, c = \underline{} \end{aligned}$
::a 14. a=6,b=8,c

15. $\begin{aligned} a = 9, b = 12, c = \underline{} \end{aligned}$
::15.a=9,b=12,c=15,b=12,c

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

Indirect Measurement ::间接计量

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Review ::回顾

Review (Answers) ::回顾(答复)

Indirect Measurement
::间接计量

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Review
::回顾

Review (Answers)
::回顾(答复)