Course: 10.10 利用毕达哥里定理寻找距离-interactive

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使用毕达哥里定理寻找距离

The Pythagorean Theorem in the Modern World
::现代世界的毕达哥里安理论论

If you want to know how the Pythagorean Theorem has shaped the world around you, look at a cell phone. Your cell phone receives a signal through a process from cell phone towers using trigonometric principles derived from the Pythagorean Theorem.
::如果您想知道毕达哥里安理论如何塑造你周围的世界, 请看一看手机。您的手机通过手机塔的程序接收信号, 使用源自毕达哥里安理论的三角测量原理。

Use the interactive below to explore how this is done.
::使用下面的交互方式来探讨如何做到这一点。

INTERACTIVE

Finding Distance Using the Pythagorean Theorem

Some protected animals have tracking chips that help biologists study them in their environments.

::一些受保护的动物有跟踪芯片帮助生物学家在它们的环境中研究它们

Click the buttons to turn on the cell towers.
::单击按钮打开单元格塔。

Can you find how far away is the Rocky Mountain elk?
::你能找到落基山有多远吗?

ythagorean-theorem-and-its-converse" quiz-url="https://www.ck12.org/assessment/ui/embed.html?test/view/5f443f1f5a6ee7300df4e956&collectionHandle=geometry&collectionCreatorID=3&conceptCollectionHandle=geometry-:

ythagorean-theorem-and-its-converse&mode=lite" test-id="5f443f1f5a6ee7300df4e956">

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As shown in the interactive, the base angles of the triangle are 60 degrees. Find measure of the third angle.
::如交互式边框所示,三角形的基角为60度。查找第三个角的度量。

The Shortest Distance From One Place to Another
::从一个地方到另一个地方最短距离

The shortest distance between any two points is a straight line . This concept is analyzed in great detail by applied mathematicians who are hired by transportation companies to design the most efficient routes possible. These mathematicians need to find the best route to minimize both distance and time. The Pythagorean Theorem is one of many tools at their disposal to accomplish this. A school district needs to find the route with the shortest distance for a school bus to get from the school to the first house on its route. Use the interactive below to explore and compare the different options.
::任何两个点之间的最短距离是一条直线。由运输公司雇用的实用数学家对这一概念进行了非常详细的分析,这些数学家设计了尽可能高效的路线。这些数学家需要找到最佳路线,以尽量减少距离和时间。 Pytagorian Theorem是他们可用于实现这一目标的多种工具之一。一个学区需要找到一条距离最短的路线,以便乘坐校车从学校到路线上的第一栋房子。使用下面的交互式方式来探索和比较不同的选择。

INTERACTIVE

What Is The Shortest Distance?

Which of the three routes do you think has the shortest distance?
::你认为这三条路线中哪条距离最短?

Click the button in order to switch between the three routes.
::单击此按钮可在三条路径之间切换。

ythagorean-theorem-and-its-converse" quiz-url="https://www.ck12.org/assessment/ui/embed.html?test/view/5f443fce5ca058f5baccdd3f&collectionHandle=geometry&collectionCreatorID=3&conceptCollectionHandle=geometry-:

ythagorean-theorem-and-its-converse&mode=lite" test-id="5f443fce5ca058f5baccdd3f">

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Choose the TRUE statements.
::选择 TRUE 语句。

Using the Pythagorean Theorem to Find Distance
::使用毕达哥里定理来寻找距离

Many apps on your cell phone also use the Pythagorean Theorem. A mapping app uses the coordinates of latitude and longitude to turn the Earth into a spherical coordinate plane . As you learned in the previous activity, the distance between two points could be found using a straight line. When points line up along an axis such as the same latitude line or the same y-axis line, it is easy to find the distance. Use the interactive below to explore this idea:
::您手机上的许多应用程序也使用 Pythagorean Theorem 。映射应用程序使用纬度和经度坐标将地球变成球形坐标平面。正如您在前一次活动中学到的, 使用直线可以找到两个点之间的距离。当点沿着同一纬度线或相同的 y- 轴线等轴线排行时, 很容易找到距离。使用下面的互动来探索这个想法 :

INTERACTIVE

Using Pythagorean Theorem to Find Distance

Move the two, red points and try to find the distance between them.
::移动两个红点, 并尝试找到它们之间的距离。

Press "Hint" to see the hypotenuse of a right triangle formed by the points.
::按下“ 提示” 来查看由点组成的右三角形的下限值。

ythagorean-theorem-and-its-converse" quiz-url="https://www.ck12.org/assessment/ui/embed.html?test/view/5f4440273745846eb53cfd57&collectionHandle=geometry&collectionCreatorID=3&conceptCollectionHandle=geometry-:

ythagorean-theorem-and-its-converse&mode=lite" test-id="5f4440273745846eb53cfd57">

Now that you know how to use the coordinates of objects to find the distance. E xamine this on a map.
::现在你知道如何使用天体坐标找到距离了。在地图上检查一下。

CK-12 PLIX Interactive: Neighborhood Map
::CK-12 PLIX 互动:邻居地图

Discussion Question
::讨论问题

Develop an algorithm for finding the distance of two points on a coordinate plane. What are the steps you followed? Can you write it as a formula?
::开发一个算法, 以查找坐标平面上两点的距离。您遵循的步骤是什么 ? 您能把它写成公式吗 ?

Distance Formula
::距离公式

Use the interactive below to help understand the ideas behind the distance formula.
::使用下面的互动来帮助理解距离公式背后的想法。

CK-12 PLIX Interactive: Right Triangles
::CK-12 PLIX 互动:右三角

The distance formula is a formula that is used to measure distances on a coordinate plane. However, it’s just the Pythagorean Theorem rearranged. If you told someone how to solve for $\begin{aligned} c \end{aligned}$ , you would be saying the formula exactly:
::距离公式是用来测量坐标平面距离的公式。然而,这只是Pytagoren Theorem的重新排列。如果你告诉别人如何解决c, 你会说公式的准确性:

Find the value of $\begin{aligned} a \end{aligned}$ and $\begin{aligned} b \end{aligned}$ .
::查找 a 和 b 的值。

Square $\begin{aligned} a \end{aligned}$ and $\begin{aligned} b \end{aligned}$ and add the sum.
::a和b方形并加上总和。

Take the square root.
::拿平方根来

The distance formula is essentially the algorithm above, but written using numbers and operations.
::距离公式基本上是上面的算法,但用数字和操作来写。

Find the value of $\begin{aligned} a \end{aligned}$ and $\begin{aligned} b \end{aligned}$ . The value of $\begin{aligned} a \end{aligned}$ can be determined by taking the difference of the x-coordinates $\begin{aligned} | x_{2} - x_{1} | \end{aligned}$ . The value of $\begin{aligned} b \end{aligned}$ can be determined by taking the difference of the y-coordinates $\begin{aligned} | y_{2} - y_{1} | \end{aligned}$
::查找 a 和 b 的值。 a 的值可以通过取取 x 坐标 {x2 - x1\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

Square a and b and add the sum. $\begin{aligned} {| x_{2} - x_{1} |}^{2} + {| y_{2} - y_{1} |}^{2} \end{aligned}$
::平方 a 和 b , 并加上和数。 @ x2- x1 @ @ 2\ y2-y1 @ 2

Take the square root. $\begin{aligned} \sqrt{{| x_{2} - x_{1} |}^{2} + {| y_{2} - y_{1} |}^{2}} \end{aligned}$
::拿平方根。 * _x2 -x1 -2\y2 -y1\2

Example
::示例示例示例示例

Use the distance formula to find the distance between the points (3, 6) and (8, -1).
::使用距离公式查找点(3,6)和点(8,1)之间的距离。

Let’s begin by identifying our values:
::让我们首先确定我们的价值观:

$\begin{aligned} x_{1} = 3 \end{aligned}$
::x1=3

$\begin{aligned} x_{2} = 8 \end{aligned}$
::x2=8

$\begin{aligned} y_{1} = 6 \end{aligned}$
::y1=6 y1=6

$\begin{aligned} y_{2} = - 1 \end{aligned}$
::y2%1

Next substitute these values into the formula.
::然后将这些值替换为公式。

$\begin{aligned} \sqrt{{| 8 - 3 |}^{2} + {| - 1 - 6 |}^{2}} \end{aligned}$

From here you can simplify using the order of operations and PEMDAS.
::在此您可以使用操作顺序和 PEMDAS 来简化操作。

$\begin{aligned} \sqrt{{| 8 - 3 |}^{2} + {| - 1 - 6 |}^{2}} \\ \sqrt{{| 5 |}^{2} + {| - 7 |}^{2}} \\ \sqrt{5^{2} + 7^{2}} \\ \sqrt{25 + 49} \\ \sqrt{74} \end{aligned}$

Answer: The prime factorization of 74 are $\begin{aligned} 2 \cdot 37 \end{aligned}$ so you cannot simplify it, but you can approximate it to 8.6.
::回答:74的基数乘数是237 这样你就不能简化它,但你可以把它接近8.6。

Summary

To find the distance between the points $\begin{aligned} (x_{1}, y_{1}) \end{aligned}$ and $\begin{aligned} (x_{2}, y_{2}), \end{aligned}$ use the equation: $\begin{aligned} \sqrt{| x_{2} - x_{1} |^{2} + | y_{2} - y_{1} |^{2}} \end{aligned}$
::要找到点( x1,y1) 和点( x2,y2) 之间的距离, 请使用方程式 : @ x2 - x1, @ @ @ 2\ y2 -y1} 2

Section outline

General

使用毕达哥里定理寻找距离

The Pythagorean Theorem in the Modern World ::现代世界的毕达哥里安理论论

The Shortest Distance From One Place to Another ::从一个地方到另一个地方最短距离

Using the Pythagorean Theorem to Find Distance ::使用毕达哥里定理来寻找距离

CK-12 PLIX Interactive: Neighborhood Map ::CK-12 PLIX 互动:邻居地图

Distance Formula ::距离公式

CK-12 PLIX Interactive: Right Triangles ::CK-12 PLIX 互动:右三角

The Pythagorean Theorem in the Modern World
::现代世界的毕达哥里安理论论

The Shortest Distance From One Place to Another
::从一个地方到另一个地方最短距离

Using the Pythagorean Theorem to Find Distance
::使用毕达哥里定理来寻找距离

CK-12 PLIX Interactive: Neighborhood Map
::CK-12 PLIX 互动:邻居地图

Distance Formula
::距离公式

CK-12 PLIX Interactive: Right Triangles
::CK-12 PLIX 互动:右三角