Section: 未知三角形的尺寸 | 10.4未知三角形的尺寸

Section outline

Marie has a collection of photos of buildings shaped like pyramids . As she looked through the photos, she wondered how tall one of the buildings was. She knows that the area of the triangle shaped surface is 900 square feet. She also knows that the length of each side of the base of the pyramid (which is also the length of the base of each triangular side) is 60 feet. Given the base and area of the triangles forming the pyramid, how can Marie determine the height of the pyramid?
::Marie收集了像金字塔一样的建筑物照片。她从照片中看到,她想知道其中一座建筑物有多高。她知道三角形表面的面积是900平方英尺。她还知道金字塔底部每一边的长度(也是三角形底部的长度)是60英尺。鉴于形成金字塔的三角的底部和面积,Marie如何确定金字塔的高度?

In this concept, you will learn how to find the unknown dimensions of triangles given the area and another dimension.
::在此概念中, 您将学习如何找到三角形的未知维度, 以区域和另一个维度为条件。

Finding Unknown Dimensions of Triangles
::查找三角形的未知尺寸

There may be a situation when you will be given the area of a triangle and one other dimension and asked to find the missing dimension. When this happens, you can figure out the missing dimension by using the formula for the area of a triangle.
::可能会出现这样的情况:您将获得三角形和另一个维度的区域, 并被要求查找缺失的维度。当这种情况发生时, 您可以使用三角形区域的公式来解析缺失的维度。

$\begin{aligned} A = \frac{1}{2} b h \end{aligned}$

If you have been given the area and the base of a triangle, then you can use the formula to figure out the height. If you have been given the area and the height then you can figure out the base.
::如果给定了三角形的面积和基底,那么您就可以使用公式来计算高度。如果给定了面积和高度,那么您就可以找出基数。

Now let’s use the formula to find the base of a triangle when the area and height are given.
::现在让我们使用公式在给定面积和高度时找到三角形的底部。

A triangle has an area of 36 square inches and a height of 6 inches. What is the measure of the base?
::三角形的面积为36平方英寸,高度为6英寸。基数是多少?

Start by looking at the formula for finding the area of a triangle.
::首先从寻找三角形区域的公式开始。

$\begin{aligned} A = \frac{1}{2} b h \end{aligned}$

First, substitute the given information into the formula.
::首先,将给定信息替换为公式。

$\begin{aligned} 36 = \frac{1}{2} (6) b \end{aligned}$

Next, multiply one-half by six.
::下一个乘以半乘以六

$\begin{aligned} 36 = 3 b \end{aligned}$

Then, solve for the base by dividing both sides by 3.
::然后,通过将双方除以3来解决基地问题。

$\begin{aligned} \begin{array}{rcl} \frac{36}{3} & = & \frac{3 b}{3} \\ 12 & = & b \end{array} \end{aligned}$

The answer is 12. The base of this triangle is 12 inches.
::答案是12 这个三角的底部是12英寸

Now, let’s find the height of a triangle with a given area and base.
::现在,让我们找到三角形的高度,加上给定的面积和底部。

A triangle has an area of 64 square inches and a base of 8 inches. What is the measure of the base?
::三角形的面积为64平方英寸,基数为8英寸。基数是多少?

To figure this out, start by looking at the formula for finding the area of a triangle.
::要弄清楚这一点,首先从寻找三角形区域的公式开始。

$\begin{aligned} A = \frac{1}{2} b h \end{aligned}$

First, substitute the given information into the formula.
::首先,将给定信息替换为公式。

$\begin{aligned} 64 = \frac{1}{2} (8) h \end{aligned}$

Next, multiply one-half by 8.
::下一个乘以一半乘以八

$\begin{aligned} 64 = 4 h \end{aligned}$

Then, solve for the base by dividing both sides by 4.
::然后,通过将双方除以4来解决基地问题。

$\begin{aligned} \begin{array}{rcl} \frac{64}{4} & = & \frac{4 h}{4} \\ 16 & = & h \end{array} \end{aligned}$

The answer is 16. The height of this triangle is 16 inches.
::答案是16 这个三角形的高度是16英寸

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Marie and her photos of pyramid buildings.
::之前,你得到一个问题关于玛丽和她的照片的金字塔建筑。

Marie found out the area of each triangle shaped surface is 900 square feet. She also knows that the length of each side of the base of the pyramid (which is also the length of the base of each triangular side) is 60 feet. Marie found the formula for computing the area of a triangle, and decided to use it to figure out the triangle’s height.
::Marie发现每个三角形表面的面积为900平方英尺。她也知道金字塔底部的每侧长度(也是每三角形底部的长度)为60英尺。 Marie找到了计算三角形面积的公式,并决定用它来计算三角形的高度。

$\begin{aligned} A = \frac{1}{2} b h \end{aligned}$

First, she should substitute the given information into the formula.
::首先,她应把所提供的信息改成公式。

$\begin{aligned} 900 = \frac{1}{2} (60) h \end{aligned}$

Next, multiply one-half by 60.
::下一个乘以一半乘以60

$\begin{aligned} 900 = 30 h \end{aligned}$

Then, solve for the base by dividing both sides by 30.
::然后,通过将双方除以30来解决基地问题。

$\begin{aligned} \begin{array}{rcl} \frac{900}{30} & = & \frac{30 h}{30} \\ 30 & = & h \end{array} \end{aligned}$

The answer is 30. The height of this pyramid is 30 feet.
::答案是30 这个金字塔的高度是30英尺

Example 2
::例2

Find the missing dimension of the following triangle.
::查找以下三角形的缺失维度。

The area of the triangle is 48 square feet. The base of the triangle is 12 feet. What is the height?
::三角形的面积是48平方英尺。三角形的底部是12英尺。高度是多少?

Start by looking at the formula for finding the area of a triangle.
::首先从寻找三角形区域的公式开始。

$\begin{aligned} A = \frac{1}{2} b h \end{aligned}$

First, fill in the given information.
::首先,填写给定信息。

$\begin{aligned} 48 = \frac{1}{2} (12) h \end{aligned}$

Next, multiply one-half by 12.
::下一个乘以一半乘以十二乘以十二

$\begin{aligned} 48 = 6 h \end{aligned}$

Then, solve for the base by dividing 48 by 6.
::然后,通过将48除以6 来解决基地问题。

$\begin{aligned} \begin{array}{rcl} \frac{48}{6} & = & \frac{6 h}{6} \\ 8 & = & h \end{array} \end{aligned}$

The answer is 8. The height of this triangle is 8 feet.
::答案是8 这个三角形的高度是8英尺

Example 3
::例3

A triangle has an area of 42 sq. ft. If the base is 12 feet, what is the measure of the height?
::三角形的面积为42平方米方位。如果基数是12英尺,那么高度的量度是多少?

To figure this out, start by looking at the formula for finding the area of a triangle.
::要弄清楚这一点,首先从寻找三角形区域的公式开始。

$\begin{aligned} A = \frac{1}{2} b h \end{aligned}$

First, substitute the given information into the formula.
::首先,将给定信息替换为公式。

$\begin{aligned} 42 = \frac{1}{2} (12) h \end{aligned}$

Next, multiply one-half by 12.
::下一个乘以一半乘以十二乘以十二

$\begin{aligned} 42 = 6 h \end{aligned}$

Then, solve for the base by dividing both sides by 6.
::然后,通过将双方除以6来解决基地问题。

$\begin{aligned} \begin{array}{rcl} \frac{42}{6} & = & \frac{6 h}{6} \\ 7 & = & h \end{array} \end{aligned}$

The answer is 7. The height of this triangle is 7 feet.
::答案是7 这个三角形的高度是7英尺

Example 4
::例4

A triangle has an area of 16 sq. cm. If the height of the triangle is 4 cm, what is the measure of the base?
::三角形的面积为 16 sq. cm。如果三角形的高度为 4 cm, 则基数的测量值是多少 ?

To figure this out, start by looking at the formula for finding the area of a triangle.
::要弄清楚这一点,首先从寻找三角形区域的公式开始。

$\begin{aligned} A = \frac{1}{2} b h \end{aligned}$

First, substitute the given information into the formula.
::首先,将给定信息替换为公式。

$\begin{aligned} 16 = \frac{1}{2} b (4) \end{aligned}$

Next, multiply one-half by 4.
::下一个乘以一半乘以四

$\begin{aligned} 16 = 2 b \end{aligned}$

Then, solve for the base by dividing both sides by 2.
::然后,通过将双方除以2来解决基地问题。

$\begin{aligned} \begin{array}{rcl} \frac{16}{2} & = & \frac{2 b}{2} \\ 8 & = & b \end{array} \end{aligned}$

The answer is 8. The height of this triangle is 8 cm.
::答案是8 这个三角形的高度是8厘米

Example 5
::例5

A triangle has an area of 40 sq. meters and a base of 10. What is the height?
::三角形的面积为40平方米,基数为10米,高度是多少?

Start by looking at the formula for finding the area of a triangle.
::首先从寻找三角形区域的公式开始。

$\begin{aligned} A = \frac{1}{2} b h \end{aligned}$

First, substitute the given information into the formula.
::首先,将给定信息替换为公式。

$\begin{aligned} 40 = \frac{1}{2} (10) h \end{aligned}$

Next, multiply one-half by 10.
::下一个乘以半乘10乘10

$\begin{aligned} 40 = 5 h \end{aligned}$

Then, solve for the base by dividing both sides by 5.
::然后,通过将双方除以5来解决基地问题。

$\begin{aligned} \begin{array}{rcl} \frac{40}{5} & = & \frac{5 h}{5} \\ 8 & = & h \end{array} \end{aligned}$

The answer is 8. The height of this triangle is 8 meters.

::答案是8 这个三角形的高度是8米

Review
::回顾

Find the missing dimension given the area and either the base or height of a triangle.
::查找根据区域以及三角形的底部或高度而发现的缺失的维度。

Area = 15 sq. in, Base = 10 in, what is the height?
::面积=15平方米,基地=10英寸,高度是多少?

Area = 40 sq. in, Base = 20 in, what is the height?
::面积=40平方米,基地=20平方米,高度是多少?

Area = 24 sq. ft, Base = 8 ft, what is the height?
::面积=24平方英尺,基地=8平方英尺,高度是多少?

Area = 16 sq. m, Base = 8 m, what is the height?
::面积=16平方米,基地=8米,高度是多少?

Area = 25 sq. in, height = 5 in, what is the base?
::面积 = 25 平方英寸,高度 = 5 英寸,基数是多少?

Area = 36 sq. ft, Height = 6 ft, what is the base?
::面积=36平方英尺,高度=6平方英尺,基数是多少?

Area = 54 sq. cm, Height = 9 cm, what is the base?
::面积=54平方公里,高度=9厘米,基数是多少?

Area = 80 sq. meters, Base = 16 meters, what is the height?
::面积=80平方米,基地=16米,高度是多少?

Area = 75 sq. meters, Base = 10 meters, what is the height?
::面积=75平方米,基地=10米,高度是多少?

Area = 90 sq. meters, Base = 30 meters, what is the height?
::面积=90平方米,基地=30米,高度是多少?

Area = 180 sq. meters, Base = 10 meters, what is the height?
::面积=180平方米,基地=10米,高度是多少?

Area = 90 sq. meters, Base = 15 meters, what is the height?
::面积=90平方米,基地=15米,高度是多少?

Area = 120 sq. meters, Base = 60 meters, what is the height?
::面积=120平方米,基地=60米,高度是多少?

Area = 150 sq. meters, Base = 50 meters, what is the height?
::面积=150平方米,基地=50米,高度是多少?

Area = 280 sq. meters, Base = 140 meters, what is the height?
::面积=280平方米,基地=140米,高度是多少?

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

Finding Unknown Dimensions of Triangles ::查找三角形的未知尺寸

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

Finding Unknown Dimensions of Triangles
::查找三角形的未知尺寸

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