Section: 毕达哥里安神话及其对话 | 9.7 毕达哥里安神论及其交汇

Section outline

Last year, Yuli and his dad made a square bed for vegetables and edged it with timbers. This year his mom wants another one for flowers that measures 5 feet by 5 feet and has a timber diagonally through the middle to separate the sections. How long should the dividing timber be?
::去年,尤利和他爸爸做了一张蔬菜的平方床,用木头围住它。今年,他妈妈想要一张5英尺乘5英尺的花朵,中间有一根木头对角,把两块分割开来。分隔的木头应该多长?

In this concept, you will learn how to use the Pythagorean Theorem and its converse .
::在这个概念中,你将学会如何使用毕达哥里安理论及其对白。

Using the Pythagorean Theorem and its Converse
::使用毕达哥里安定理词及其对立词

You already know that a square consists of four equal sides and four equal, right, angles.
::你已经知道一个广场由四个平等面组成四个平等,权利,角度。

When you divide a square in half by a diagonal, it forms two right triangles.
::当用对角法将一个正方块除以一半时,它会形成两个右三角形。

In the labeled right triangle below:
::在以下标记的右三角形中:

Sides $\begin{aligned} a \end{aligned}$ and $\begin{aligned} b \end{aligned}$ are adjacent to the right angle and are called the legs of the triangle.
::a和b的侧面与右角相邻,称为三角形的腿。

Side $\begin{aligned} c \end{aligned}$ , the longest side, is opposite the right angle and is called the hypotenuse of the triangle.
::侧面 c, 最长的侧面, 右角度对面, 被称为三角形的下限。

The Pythagorean Theorem is named for the Greek mathematician Pythagoras, who discovered the unique relationship between the dimensions of a right triangle. The theorem says that the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse.
::Pytagoren Theorem的名字是希腊数学家Pytagoras的名字,他发现了右三角形维度之间的独特关系。理论家说右三角形两腿平方的和等于下角。

$\begin{aligned} a^{2} + b^{2} = c^{2} \end{aligned}$

::a2+b2=c2 = c2 =

Let’s look at an example.
::让我们举个例子。

Find the hypotenuse.
::找到低温剂

First, write the formula , and substitute what you know.
::首先,写出公式, 并取代你知道的东西。

$\begin{aligned} \begin{array}{rcl} a^{2} + b^{2} & = & c^{2} \\ 3^{2} + 4^{2} & = & c^{2} \end{array} \end{aligned}$

::a2+b2=c232+42=c2

Next, do the calculations.
::接下来,做计算。

$\begin{aligned} \begin{array}{rcl} 9 + 16 & = & c^{2} \\ 25 & = & c^{2} \end{array} \end{aligned}$

::9+16=c225=c2

Then, take the square root of both sides of the equation .
::然后,从方程式两侧的平方根中取出平方根。

$\begin{aligned} 5 = c \end{aligned}$

::5=c 5=c

The answer is 5. The hypotenuse of the triangle is 5.
::答案是5. 三角形的下限值是5。

Here’s another example.
::以下是另一个例子。

Find the length of the hypotenuse in the triangle below by using the Pythagorean Theorem.
::使用 Pytagoren 理论来查找下方三角形的下方值长度。

First, write the formula, and substitute.
::首先,写出配方,替换

$\begin{aligned} \begin{array}{rcl} c^{2} & = & a^{2} + b^{2} \\ c^{2} & = & 6^{2} + 8^{2} \end{array} \end{aligned}$

::c2=a2+b2c2=62+82

Next, calculate the squares.
::接下来,计算方形。

$\begin{aligned} \begin{array}{rcl} c^{2} & = & 36 + 64 \\ c^{2} & = & 100 \end{array} \end{aligned}$

::c2=36+64c2=100

Then take the square root of each side.
::那就把每一方的平方根取下来

The answer is $\begin{aligned} c = 10 \end{aligned}$ . The length of the hypotenuse is 10.
::答案是C=10。下限长度为10。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Yuli and his garden beds.
::早些时候,你听说了尤利和他的花园床有问题

Yuli needs to know how long a diagonal timber he needs for a $\begin{aligned} 5 f t \times 5 f t \end{aligned}$ square.
::尤利需要知道 5英尺5英尺5英尺方形的对角木需要多久

First, use the Pythagorean Theorem and substitute in the values you know.
::首先,使用毕达哥里安神话, 并取代你所知道的价值观。

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

::c2=a2+b2c2=52+52

Next, calculate the value of the squares.
::接下来,计算方形的值。

$\begin{aligned} \begin{array}{rcl} c^{2} & = & 25 + 25 \\ c^{2} & = & 50 \end{array} \end{aligned}$

::c2=25+25c2=50

Then take the square root of each side.
::那就把每一方的平方根取下来

$\begin{aligned} \sqrt{c^{2}} = \sqrt{50} \end{aligned}$

::c2=50 (c2=50)

Use your calculator.
::使用你的计算器。

$\begin{aligned} c = 7.071067812 \dots \end{aligned}$

::=7.071067812... =7.071067812...

Round the number.
::圆圆数字。

$\begin{aligned} c \approx 7.07 \end{aligned}$

::c7.07

The answer is that the hypotenuse $\begin{aligned} c \approx 7.07 \end{aligned}$ , but since Yuli may have a hard time measuring $\begin{aligned} \frac{7}{100} \end{aligned}$ of a foot, it is safe to say that Yuli can use a 7 ft timber.
::答案是,下层的温度是7.07, 但是由于尤利可能很难过, 每英尺7100英尺, 可以说尤利可以使用7英尺的木材。

Example 2
::例2

What is the length of the hypotenuse of this triangle?
::这个三角形的下限长度是多少?

First, write the formula and fill in what you know.
::首先,写出公式,填写你所知道的内容。

$\begin{aligned} 7^{2} + 10^{2} = c^{2} \end{aligned}$

::72+102=c2

Next, do the calculations.
::接下来,做计算。

$\begin{aligned} \begin{array}{rcl} 49 + 100 & = & c^{2} \\ 149 & = & c^{2} \end{array} \end{aligned}$

::49+100=c249=c2

Then take the square roots.
::那就从平方根开始吧

$\begin{aligned} \sqrt{149} = c \end{aligned}$

::149=c

Since 149 is not a perfect square , use your calculator.
::149不是一个完美的广场,使用你的计算器。

$\begin{aligned} \sqrt{149} = 12.20655562 \dots \end{aligned}$

Round to the nearest tenth.
::圆到最近的十点。

The answer is $\begin{aligned} \sqrt{149} = 12.2 \end{aligned}$
::答案是149=12.2

Example 3
::例3

A right triangle has legs that measure 9 and 12.
::右三角有9和12的足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足足。

What is the hypotenuse?
::什么是低温?

First, fill the values into the equation.
::首先,将数值填入方程。

$\begin{aligned} 9^{2} + 12^{2} = c^{2} \end{aligned}$

::92+122=c2

Next, perform the calculations.
::接下来,进行计算。

$\begin{aligned} \begin{array}{rcl} 81 + 144 & = & c^{2} \\ 225 & = & c^{2} \end{array} \end{aligned}$

::81+144=c2225=c2

Then, determine the square roots.
::然后,决定平方根。

$\begin{aligned} 15 = c \end{aligned}$

::15=c 15=c

The answer is 15. The hypotenuse is 15.
::答案是15,低温是15

Example 4
::例4

Find the hypotenuse of a right triangle with legs of 3 and 6.
::找到右三角形的下限,右三角形的腿为3和6。

$\begin{aligned} \begin{array}{rcl} a^{2} + b^{2} = c^{2} \\ 3^{2} + 6^{2} = c^{2} \end{array} \end{aligned}$

::a2+b2=c232+62=c2

Next, square the legs.
::下一位,把腿平开

$\begin{aligned} \begin{array}{rcl} 9 + 36 = c^{2} \\ 45 = c^{2} \end{array} \end{aligned}$

::9+36=c245=c2

Then calculate the square roots of both sides of the equation.
::然后计算方程两侧的平方根。

$\begin{aligned} \sqrt{45} = c \end{aligned}$

::45=c=45=c

Use a calculator.
::使用计算器。

$\begin{aligned} c = 6.708203933 \dots \end{aligned}$

::=6.708203933... =6.708203933...

The answer is $\begin{aligned} c \approx 6.7 \end{aligned}$ . The hypotenuse is approximately 6.7.
::答案是C6.7,低温大约为6.7。

Example 5
::例5

Claire’s mom needs to fix a broken shingle on their house. If the shingle is 16 feet above the ground and Claire places the foot of the ladder 12 feet from the wall, how long will the ladder need to be?
::Claire的母亲需要修复他们家的破碎闪光灯。如果闪光灯在地上16英尺高,而Claire把梯子脚从墙上移到距离墙12英尺的地方,梯子需要多久?

First, draw a picture to help determine what you are solving for.
::首先,绘制一张图片,帮助确定你正在解决的是什么。

In this picture, you can see that the ladder is the hypotenuse.
::在这张图中,你可以看到梯子是低温的。

Next, insert the values of the legs into the formula for the Pythagorean Theorem.
::下一步,将腿的值插入毕达哥伦理论的公式中。

$\begin{aligned} \begin{array}{rcl} c^{2} & = & a^{2} + b^{2} \\ c^{2} & = & 12^{2} + 16^{2} \end{array} \end{aligned}$

::c2=a2+b2c2=122+162

Next, calculate the value of the squares.
::接下来,计算方形的值。

$\begin{aligned} \begin{array}{rcl} c^{2} & = & 144 + 256 \\ c^{2} & = & 400 \end{array} \end{aligned}$

::c2=144+256c2=400

Then take the square root of each side.
::那就把每一方的平方根取下来

The answer is $\begin{aligned} c = 20 \end{aligned}$ . The length of the hypotenuse is 10, and the ladder needs to be 10 feet long.
::答案是 C=20 。下限长度为 10 , 梯子需要 10 英尺长。

Review
::回顾

Use the Pythagorean Theorem to find the length of side $\begin{aligned} c \end{aligned}$ , the hypotenuse. You may round to the nearest tenths place when necessary.
::使用 Pythagorean 定理器查找 c 边的长度, 下限。必要时您可以圆到最近的十分之一处。

$\begin{aligned} a = 6, b = 8, c = ? \end{aligned}$
::a=6,b=8,c=?

$\begin{aligned} a = 9, b = 12, c = ? \end{aligned}$
::a=9,b=12,c=9,b=12,c=9,b=12?

$\begin{aligned} a = 12, b = 16, c = ? \end{aligned}$
::a=12,b=16,c=12,b=16,c=16,c=12,b=16,b=16,c=16,c=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,c=16,c=16,c=16,c=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,b=16,c=16,c=16,b=16,b=16,b=16,c=16,b=16,b=16,b=16,b=16,b=16,b=16,c=16,b=16,b=16,b=16,b=16,c=16,b=16,b=16,b=16,b=16,b=16,b=16,b=

$\begin{aligned} a = 15, b = 20, c = ? \end{aligned}$
::a=15,b=20,c=?

$\begin{aligned} a = 18, b = 24, c = ? \end{aligned}$
::a=18,b=24,c=?

$\begin{aligned} a = 24, b = 32, c = ? \end{aligned}$
::=24,b=32,c=? =24,b=32,c=?

$\begin{aligned} a = 5, b = 7, c = ? \end{aligned}$
::a=5,b=7,c=?

$\begin{aligned} a = 9, b = 11, c = ? \end{aligned}$
::a=9,b=11,c=9,b=11,c=9,b=11,c=9,b=11,c=9,b=11,b=11,c=9,b=9,b=11,c=9,b=9,b=11,b=11,c=9,b=9,b=11,b=11,c=9,b=9,b=11,b=11,c=9,b=9,b=9,b=11,b=11,c=9,b=9,b=11,c=9,b=9,b=11,b=11,c=9,b=9,b=11,c=9,b=9,b=9,b=11,c=9,b=9,b=9,b=11,c=?

$\begin{aligned} a = 10, b = 12, c = ? \end{aligned}$
::=10,b=12,c=? =10,b=12,c=?

$\begin{aligned} a = 12, b = 8, c = ? \end{aligned}$
::a=12,b=8,c=?

$\begin{aligned} a = 11, b = 8, c = ? \end{aligned}$
::a=11,b=8,c=?

$\begin{aligned} a = 9, b = 8, c = ? \end{aligned}$
::a=9,b=8,c=9,b=8,c=8?

$\begin{aligned} a = 15, b = 7, c = ? \end{aligned}$
::a=15,b=7,c=?

$\begin{aligned} a = 14, b = 3, c = ? \end{aligned}$
::a=14,b=3,c=?

$\begin{aligned} a = 12, b = 8, c = ? \end{aligned}$
::a=12,b=8,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。

Resources
::资源

Section outline

Using the Pythagorean Theorem and its Converse ::使用毕达哥里安定理词及其对立词

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Resources ::资源

Using the Pythagorean Theorem and its Converse
::使用毕达哥里安定理词及其对立词

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)

Resources
::资源