课程： 11.8 毕达哥里安神论及其交汇

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选择节毕达哥里安神话及其对话

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毕达哥里安神话及其对话
Pythagorean Theorem and its Converse
::毕达哥里安神话及其对话

Teresa wants to string a clothesline across her backyard, from one corner to the opposite corner. If the yard measures 22 feet by 34 feet, how many feet of clothesline does she need?
::Teresa想在她的后院,从一个角落到另一个角落,绑一条衣绳。如果院子用22英尺乘34英尺,她需要多少英尺的衣绳?

The Pythagorean Theorem is a statement of how the lengths of the sides of a right triangle are related to each other. A right triangle is one that contains a 90 degree angle. The side of the triangle opposite the 90 degree angle is called the hypotenuse and the sides of the triangle adjacent to the 90 degree angle are called the legs .
::Pytagorean Theorem 是一个关于右三角形两侧的长度如何互相关联的说明。右三角形是一个包含90度角的三角形。 90度角对面的三角形的侧面称为下角, 90度角对面的三角形的侧面称为腿。

If we let $\begin{aligned} a \end{aligned}$ and $\begin{aligned} b \end{aligned}$ represent the legs of the right triangle and $\begin{aligned} c \end{aligned}$ represent the hypotenuse then the Pythagorean Theorem can be stated as:
::如果我们让 a 和 b 代表右三角形的腿, c 代表下方, 那么毕达哥里安理论可以被说成:

In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. That is: $\begin{aligned} a^{2} + b^{2} = c^{2} \end{aligned}$ .
::在右三角形中,下角长度的平方等于腿长度的平方和。即:a2+b2=c2。

This theorem is very useful because if we know the lengths of the legs of a right triangle , we can find the length of the hypotenuse. Also, if we know the length of the hypotenuse and the length of a leg, we can calculate the length of the missing leg of the triangle. When you use the Pythagorean Theorem, it does not matter which leg you call $\begin{aligned} a \end{aligned}$ and which leg you call $\begin{aligned} b \end{aligned}$ , but the hypotenuse is always called $\begin{aligned} c \end{aligned}$ .
::此定理非常有用, 因为如果我们知道右三角形的腿长度, 我们就可以找到下限长度。另外, 如果我们知道下限长度和一条腿长度, 我们可以计算出三角形缺失的腿长度。当您使用 Pythagorean Theorem 时, 您调用哪个腿, 您调用哪个腿并不重要, 您调用哪个腿 b, 但下限总是被称为 C 。

Although nowadays we use the Pythagorean Theorem as a statement about the relationship between distances and lengths, originally the theorem made a statement about areas. If we build squares on each side of a right triangle, the Pythagorean Theorem says that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares formed by the legs of the triangle.
::虽然现在我们用Pythagoren Theorem作为关于距离和长度之间关系的声明,但最初的理论对区域做了声明。如果我们在右三角两侧建方形,Pythagoren Theorem说,方形的侧面是下方的方形面积相当于三角形腿形成的方形面积的总和。

Use the Pythagorean Theorem and Its Converse
::使用毕达哥里定理及其交汇

The Pythagorean Theorem can be used to verify that a triangle is a right triangle. If you can show that the three sides of a triangle make the equation $\begin{aligned} a^{2} + b^{2} = c^{2} \end{aligned}$ true, then you know that the triangle is a right triangle. This is called the .
::Pytagorean 理论可以用来验证三角形是一个右三角形。如果您可以显示三角形的三边使方程式 A2+b2=c2 成为真实, 那么您就会知道三角形是一个右三角形。这被称为。

Note: When you use the Converse of the Pythagorean Theorem, you must make sure that you substitute the correct values for the legs and the hypotenuse. The hypotenuse must be the longest side. The other two sides are the legs, and the order in which you use them is not important.
::注意 : 当您使用 Pythagorean 理论的对立面时, 您必须确保您用正确的值替换腿和下限。下限必须是最长的一面。另外两边是腿, 您使用它们的顺序并不重要。

Identifying Right Triangles
::确定右三角

1. Determine if a triangle with sides 5, 12 and 13 is a right triangle.
::1. 确定侧面5、12和13的三角形是否右三角形。

The triangle is right if its sides satisfy the Pythagorean Theorem.
::如果三角形的两面满足了毕达哥里安神话三角形是对的

If it is a right triangle, the longest side has to be the hypotenuse, so we let $\begin{aligned} c = 13 \end{aligned}$ .
::如果是右三角形,最长的一面必须是下限,所以我们让C=13。

We then designate the shorter sides as $\begin{aligned} a = 5 \end{aligned}$ and $\begin{aligned} b = 12 \end{aligned}$ .
::然后,我们将较短的两边定为a=5和b=12。

We plug these values into the Pythagorean Theorem:
::我们将这些值插入毕达哥里安理论中:

$\begin{aligned} a^{2} + b^{2} = c^{2} & \Rightarrow 5^{2} + 12^{2} = c^{2} \\ 25 + 144 = 169 = c^{2} & \Rightarrow c = 13 \end{aligned}$

::a2+b2=c2=c2=52+122=c225+144=169=c2=c2=c13

The sides of the triangle satisfy the Pythagorean Theorem, thus the triangle is a right triangle.
::三角形的侧面满足了毕达哥里安理论, 因此三角形是一个右三角形。

2. Determine if a triangle with sides, $\begin{aligned} \sqrt{10}, \sqrt{15} \end{aligned}$ and 5 is a right triangle.
::2. 确定右侧三角形为10、15和5是否为右三角形。

The longest side has to be the hypotenuse, so $\begin{aligned} c = 5 \end{aligned}$ .
::最长的一面必须是下限, 所以C=5 。

We designate the shorter sides as $\begin{aligned} a = \sqrt{10} \end{aligned}$ and $\begin{aligned} b = \sqrt{15} \end{aligned}$ .
::我们指定较短的两边为a=10和b=15。

We plug these values into the Pythagorean Theorem:
::我们将这些值插入毕达哥里安理论中:

$\begin{aligned} a^{2} + b^{2} = c^{2} & \Rightarrow {(\sqrt{10})}^{2} + {(\sqrt{15})}^{2} = c^{2} \\ 10 + 15 = 25 = c^{2} & \Rightarrow c = 5 \end{aligned}$

::a2+b2=c2=c2}((10)2)+(152)=c210+15=25=c2=c2=c=5

The sides of the triangle satisfy the Pythagorean Theorem, thus the triangle is a right triangle.
::三角形的侧面满足了毕达哥里安理论, 因此三角形是一个右三角形。

Finding the Length of the Hypotenuse
::查找伪币的长度

In a right triangle one leg has length 4 and the other has length 3. Find the length of the hypotenuse.
::在右三角形上,一条腿长4,另一条长3,找出下限长度。

$\begin{aligned} Start with the Pythagorean Theorem: & a^{2} + b^{2} & = c^{2} \\ Plug in the known values of the legs: & 3^{2} + 4^{2} & = c^{2} \\ Simplify: & 9 + 16 & = c^{2} \\ 25 & = c^{2} \\ Take the square root of both sides: & c & = 5 \end{aligned}$

::以已知腿腿值: 32+42=c2 的 Pythagorean 理论: a2+b2=c2plug 开始简化: 9+16=c225=c2 将两侧的平方根: c=5 切除为平方根: c=5

Example
::示例示例示例示例

Example 1
::例1

Determine if a triangle with sides, $\begin{aligned} 2, \sqrt{21} \end{aligned}$ and 5 is a right triangle.
::确定侧边为 2,21 和 5 的三角形是否为右三角形。

The longest side has to be the hypotenuse, so $\begin{aligned} c = 5 \end{aligned}$ .
::最长的一面必须是下限, 所以C=5 。

We designate the shorter sides as $\begin{aligned} a = 2 \end{aligned}$ and $\begin{aligned} b = \sqrt{21} \end{aligned}$ .
::我们指定较短的两边为a=2和b=21。

We plug these values into the Pythagorean Theorem:
::我们将这些值插入毕达哥里安理论中:

$\begin{aligned} a^{2} + b^{2} = c^{2} & \Rightarrow {(2)}^{2} + {(\sqrt{21})}^{2} = c^{2} \\ 4 + 21 = 25 = c^{2} & \Rightarrow c = 5 \end{aligned}$

::a2+b2=c2}(2)2+(212)=c24+21=25=c2=c2=c5

The sides of the triangle satisfy the Pythagorean Theorem, thus the triangle is a right triangle.
::三角形的侧面满足了毕达哥里安理论, 因此三角形是一个右三角形。

Review
::回顾

Determine whether each set of three numbers could be the side lengths of a right triangle.
::确定每组三个数字是否可以是右三角形的侧边长度。

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

$\begin{aligned} a = 6, b = 6, c = 6 \sqrt{2} \end{aligned}$
::a=6,b=6,c=62

$\begin{aligned} a = 8, b = 8 \sqrt{3}, c = 16 \end{aligned}$
::a=8,b=83,c=16

$\begin{aligned} a = 2 \sqrt{14}, b = 5, c = 9 \end{aligned}$
::a=214,b=5,c=9

$\begin{aligned} a = 13, b = 16, c = 19 \end{aligned}$
::a=13,b=16,c=19

$\begin{aligned} a = 20, b = 99, c = 101 \end{aligned}$
::a=20,b=99,c=101

$\begin{aligned} a = 21, b = 220, c = 221 \end{aligned}$
::a=21,b=220,c=221

$\begin{aligned} a = 7, b = 2, c = \sqrt{50} \end{aligned}$
::a=7,b=2,c=50

$\begin{aligned} a = 8, b = 6, c = 10 \end{aligned}$
::a=8,b=6,c=10

$\begin{aligned} a = 7, b = \sqrt{404}, c = 25 \end{aligned}$
::a=7,b=404,c=25

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。

章节大纲

常规

毕达哥里安神话及其对话

Pythagorean Theorem and its Converse ::毕达哥里安神话及其对话

Use the Pythagorean Theorem and Its Converse ::使用毕达哥里定理及其交汇

Identifying Right Triangles ::确定右三角

Finding the Length of the Hypotenuse ::查找伪币的长度

Example ::示例示例示例示例

Example 1 ::例1

Review ::回顾

Review (Answers) ::回顾(答复)

Pythagorean Theorem and its Converse
::毕达哥里安神话及其对话

Use the Pythagorean Theorem and Its Converse
::使用毕达哥里定理及其交汇

Identifying Right Triangles
::确定右三角

Finding the Length of the Hypotenuse
::查找伪币的长度

Example
::示例示例示例示例

Example 1
::例1

Review
::回顾

Review (Answers)
::回顾(答复)