圆圈和相似

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A circle is a simple closed curve, with a set of all points at a constant distance from a fixed ( center ) point  $\begin{align*}A\end{align*}$ , in the same plane . Since the circle has only one center , you can name the circle by naming its center. In this way, you can name this as circle $\begin{align*}A\end{align*}$ .
::一个圆是一个简单封闭的曲线,与同一平面的固定(中点) A 保持恒定距离的一组所有点。由于该圆只有一个中心,您可以通过命名其中心命名圆。这样,您就可以将它命名为圆 A 。

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The radius of a circle  $\begin{align*}r\end{align*}$ , is the distance from the center of the circle to the circle. A circle's size depends only on its radius .
::圆 r 的半径是圆的中心到圆的距离。圆的大小仅取决于圆的半径。

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A similarity transformation is one or more rigid transformations ( reflection , rotation , translation) followed by a dilation. When a figure is transformed by a similarity transformation, an image is created that is similar to the original figure. In other words, two figures are similar if a similarity transformation will carry the first figure to the second figure.
::相似性变换是指一种或多种硬质变换(反思、旋转、翻译),然后是膨胀。当一个数字被相似性变换所转变时,一个图像与原始图相仿。换句话说,如果相似性变换将第一个图与第二个图相仿,两个数字是相似的。

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In the example, you will show that all circles are similar, a translation and a scale factor (from a dilation) will be found to map one circle onto another.
::在此示例中, 您将显示所有圆圈都相似, 将会找到一个翻译和一个比例系数( 来自于一个放大系数) 来将一个圆圈映射到另一个圆圈 。

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Recall two important formulas related to circles:
::回顾与圆圈有关的两个重要公式:

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• (Perimeter) of a Circle: $\begin{align*}C=2 \pi r\end{align*}$
  ::圆形( 单位) : C=2°r
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• Area of a Circle: $\begin{align*}A=\pi r^2\end{align*}$
  ::圆圆区域: Ar2

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Once you have shown that all circles are similar, you will explore how the circumferences and areas of circles are related.
::一旦你显示所有圆圈都相似,你就会探索圆圈圈圈圈圈圈圈和圆圈圈圈圈圈圈圈的关联。

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Proving Circle Similarity
::证明圆环相似性

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1. Consider circle $\begin{align*}A\end{align*}$ , centered at point $\begin{align*}A\end{align*}$ with radius $\begin{align*}r_A\end{align*}$ , and circle $\begin{align*}D\end{align*}$ , centered at point $\begin{align*}D\end{align*}$ with radius $\begin{align*}r_D\end{align*}$ . Perform a rigid transformation to bring point $\begin{align*}A\end{align*}$ to point $\begin{align*}D\end{align*}$ .
::1. 考虑以半径为rA的A点为中心、以半径为中心、以半径为RD的D点为中心的A圆。

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Draw a vector from point $\begin{align*}A\end{align*}$ to point $\begin{align*}D\end{align*}$ . Translate circle $\begin{align*}A\end{align*}$ along the vector to create circle $\begin{align*}A^\prime\end{align*}$ . Note that $\begin{align*}r_A \cong r_A^\prime\end{align*}$ .
::从 A 点到 D 点绘制矢量。 翻转圆 A 沿矢量绘制矢量以创建圆 A 。 注意 r\ r 。

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2. Dilate circle $\begin{align*}A\end{align*}$ to map it to circle $\begin{align*}D\end{align*}$ . Can you be confident that the circles are similar?
::2. 将圆 A 绘制成圆 D 。你能相信圆圈相似吗?

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Because the size of a circle is completely determined by its radius, you can use the radii to find the correct scale factor. Dilate circle $\begin{align*}A^\prime\end{align*}$ about point $\begin{align*}A^\prime\end{align*}$ by a scale factor of $\begin{align*}\frac{r_D}{r_{A^{\prime}}}\end{align*}$ .
::因为圆的大小完全取决于其半径, 您可以使用 radi 来找到正确的比例系数 。 以 rDrA_ 的 比例系数, 将 A 点的 A 点 向 A 点 向上 。

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Circle $\begin{align*}A^{\prime \prime}\end{align*}$ is the same circle as circle $\begin{align*}D\end{align*}$ . You can be confident of this because $\begin{align*}r_{A^{\prime \prime}}=\frac{r_D}{r_{A^{\prime}}} \cdot r_{A^{\prime}}=r_D\end{align*}$ and point $\begin{align*}A^{\prime \prime}\end{align*}$ is the same as point $\begin{align*}D\end{align*}$ . Because a circle is defined by its center and radius, if two circles have the same center and radius then they are the same circle.
::A 圆是和 D 圆相同的圆。 您可以对此有信心, 因为 RArDrArArD 和 A 点与 D 点相同 。 因为 圆是根据其中心或半径定义的, 如果两个圆的中间和半径相同, 那么它们就是相同的圆 。

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This means that circle $\begin{align*}A\end{align*}$ is similar to circle $\begin{align*}D\end{align*}$ , because a similarity transformation ( translation then dilation ) mapped circle $\begin{align*}A\end{align*}$ to circle $\begin{align*}D\end{align*}$ .
::这意味着圆 A 与圆 D 相似,因为相似性转换(翻译然后放大)将圆 A 绘制为圆 D 。

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Circle $\begin{align*}A\end{align*}$ and circle $\begin{align*}D\end{align*}$ were two random circles. This proves that in general, all circles are similar.
::A圆圈和D圆圈是两个随机圆圈。这证明一般来说,所有圆圈都是相似的。

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3. Show that circle $\begin{align*}A\end{align*}$ with center $\begin{align*}(-3, 4)\end{align*}$ and radius 2 is similar to circle $\begin{align*}B\end{align*}$ with center $\begin{align*}(3, 2)\end{align*}$ and radius 4.
::3. 显示圆A与中心(-3.4)和半径2相似于圆B与中心(3,2)和半径4相似。

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Translate circle $\begin{align*}A\end{align*}$ along the vector from $\begin{align*}(-3, 4)\end{align*}$ to $\begin{align*}(3, 2)\end{align*}$ . Then, dilate the image about its center by a scale factor of 2. You will have mapped circle $\begin{align*}A\end{align*}$ to circle $\begin{align*}B\end{align*}$ with a similarity transformation. This means that circle $\begin{align*}A\end{align*}$ is similar to circle $\begin{align*}B\end{align*}$ .
::将矢量的圆A从 (- 3,4) 转换为 (3, 2) 。 然后, 将其中间的图像放大为 2 比例因数 。 您将会绘制圆A 到圆B 的圆B , 并进行相似的转换。 这意味着圆A 与圆B 相似 。

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CK-12 PLIX: Translation Applications in Circle Similarity
::CK-12 PLIX: 圆圈相似的翻译应用

 

 

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Examples
::实例实例实例实例

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Example 1
::例1

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Sean has two circles, one with a radius of 1 inch and another with a radius of 3 inches.
::Sean有两个圆圈,一个圆圈半径为1英寸,另一个圆圈半径为3英寸。

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1. What is the ratio between the radii of the circles?
   ::圆圈的半径之间的比例是多少?
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2. What is the scale factor between the two circles?
   ::两个圆之间的比例因素是什么?
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3. What is the ratio between the circumferences of the circles?
   ::圆圈环形之间的比例是多少?
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4. What is the ratio between the areas of the circles?
   ::圆圈区域之间的比例是多少?
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5. What do area ratios and circumference ratios have to do with scale factor?
   ::面积比率和环绕比率与比额表系数有什么关系?

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Sean has two circles, one with a radius of 1 inch and another with a radius of 3 inches.
::Sean有两个圆圈,一个圆圈半径为1英寸,另一个圆圈半径为3英寸。

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a. What is the ratio between the radii of the circles?
::a. 圆的弧形之间的比例是多少?

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The ratio between the radii is $\begin{align*}\frac{r_A}{r_B} = \frac{3}{1}\end{align*}$ .
::弧度之间的比率是RARB=31。

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b. What is the scale factor between the two circles?
::b. 两者之间的比额表因素是什么?

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A scale factor exists because any two circles are similar. You can use the radii to determine the scale factor. The ratio between the radii is $\begin{align*}\frac{r_A}{r_B} = \frac{3}{1}\end{align*}$ so the scale factor is $\begin{align*} \frac{3}{1}=3\end{align*}$ .
::比例系数之所以存在,是因为任何两个圆圈是相似的。您可以使用弧度来确定比例系数。弧度之间的比重是 rArB=31,因此比例系数是 31=3。

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c. What is the ratio between the circumferences of the circles?
::c. 圆圈环形之间的比例是多少?

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The circumference of the smaller circle is $\begin{align*}C_B=2 \pi (1)=2 \pi\end{align*}$ . The circumference of the larger circle is $\begin{align*}C_A=2 \pi (3)=6 \pi\end{align*}$ . The ratio between the circumferences is $\begin{align*}\frac{C_A}{C_B} = \frac{6 \pi}{2 \pi}=\frac{3}{1}\end{align*}$ .
::较小圆环为CB=2(1)=2。大圆环为CA=2(3)=6。环比为CACB=6231。

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d. What is the ratio between the area of the circles?
::d. 圆圈面积之间的比例是多少?

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The area of the smaller circle is $\begin{align*}A_B=\pi(1)^2=\pi\end{align*}$ . The area of the larger circle is $\begin{align*}A_A=\pi(3)^2=9 \pi\end{align*}$ . The ratio between the areas is $\begin{align*}\frac{A_A}{A_B} = \frac{9 \pi}{\pi}=\frac{9}{1}\end{align*}$ . Note that $\begin{align*}\frac{9}{1}=\left(\frac{3}{1} \right)^2\end{align*}$ .
::较小圆的面积是 AB(1)2。大圆的面积是 AA(3)2=9。区域之间的比例是 AAAB=991。请注意,91=(312)。

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e. What do area ratios and circumference ratios have to do with scale factor?
::e. 面积比率和环绕比率与比额表系数有什么关系?

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The area ratio is the scale factor squared, because area is a two dimensional measurement. The circumference ratio is equal to the scale factor, because circumference is a one dimensional measurement.
::区域比是比例系数平方,因为区域是二维测量。环形比等于比例系数,因为环形是单维测量。

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Example 2
::例2

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Show that circle $\begin{align*}A\end{align*}$ with center $\begin{align*}(-1, 7)\end{align*}$ and radius 2 is similar to circle $\begin{align*}B\end{align*}$ with center $\begin{align*}(4, 6)\end{align*}$ and radius 3.
::显示圆 A 与 中心 (-1,7) 和 半径 2 与 中心 (4,6) 和 半径 3 的 圆 B 相似 。

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Translate circle $\begin{align*}A\end{align*}$ along the vector from $\begin{align*}(-1, 7)\end{align*}$ to $\begin{align*}(4, 6)\end{align*}$ . Then, dilate the image about its center with a scale factor of $\begin{align*}\frac{3}{2}\end{align*}$ . You will have mapped circle $\begin{align*}A\end{align*}$ to circle $\begin{align*}B\end{align*}$ with a similarity transformation. This means that circle $\begin{align*}A\end{align*}$ is similar to circle $\begin{align*}B\end{align*}$ .
::将矢量A的圆A从(-1,7)向(4,6)转换为(-1,7)至(4,6),然后将中心图象放大为32个比例因子。你将绘制圆A到圆B的圆B,并进行类似变换。这意味着圆A与圆B相似。

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Example 3
::例3

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The ratio of the circumference of circle $\begin{align*}D\end{align*}$ to the circumference of circle $\begin{align*}C\end{align*}$ is $\begin{align*}\frac{4}{3}\end{align*}$ . What is the ratio of their areas?
::D圈环绕与C圈环绕之比是43。 其面积之比是多少?

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The ratio of the circumferences is the same as the scale factor. Therefore, the scale factor is $\begin{align*}\frac{4}{3}\end{align*}$ . The ratio of the areas is the scale factor squared. Therefore, the ratio of the areas is $\begin{align*}\frac{A_1}{A_2} = \left(\frac{C_1}{C_2}\right)^2 = \left( \frac{4}{3} \right)^2=\frac{16}{9}\end{align*}$ .
::环绕比率与比重系数相同,因此,比重系数为43,区域比率为比重系数平方,因此,区域比率为A1A2=(C1C2)2=(432)=169。

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Example 4
::例4

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The ratio of the area of circle $\begin{align*}F\end{align*}$ to the area of circle $\begin{align*}E\end{align*}$ is $\begin{align*}\frac{9}{4}\end{align*}$ . What is the ratio of their radii?
::F圆区域与E圆区域之比为94,其弧度之比是多少?

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The ratio of the areas is the scale factor squared. Therefore,
$\begin{align*}\text{scale factor} = \frac{r_1}{r_2} = \sqrt{\frac{A_1}{A_2}} = \sqrt{\frac{9}{4}}=\frac{3}{2}\end{align*}$ . The ratio of the radii is the same as the scale factor, so the ratio of the radii is $\begin{align*}\frac{3}{2}\end{align*}$ .
::区域之比为比例系数平方。因此,比例系数=r1r2A1A294=32。弧度之比与比例系数相同,因此,弧度之比为32。

Remember this!

播放段落 A circle is the set of all points that are the same distance away from a specific point, called the center. ::一个圆是所有点的一组,这些点与特定点的距离相同,称为中点。 播放段落 The circumference of a circle is $\begin{align*}C=2\pi r.\end{align*}$ ::圆环环为C=2°r。 播放段落 The area of a circle is $\begin{align*}A=\pi r^2\end{align*}$ ::圆区域为 Ar2 播放段落 All circles are similar. ::所有圆圈都相似。

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Review
::审查审查审查审查

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For #1-#10, show that the circles are similar by describing the similarity transformation necessary to map one circle onto the other.
::对于 #1-#10, 显示圆圈相似, 描述为绘制一个圆形到另一个圆形所需的相似性转换 。

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1. Circle $\begin{align*}A\end{align*}$ with center $\begin{align*}(2, 7)\end{align*}$ and radius 4. Circle $\begin{align*}B\end{align*}$ with center $\begin{align*}(1, -4)\end{align*}$ and radius 3.
::1. A圆圈,中心(2,7)和半径4;B圆圈,中心(1,-4)和半径3。

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2. Circle $\begin{align*}A\end{align*}$ with center $\begin{align*}(6, 4)\end{align*}$ and radius 3. Circle $\begin{align*}B\end{align*}$ with center $\begin{align*}(-5, 6)\end{align*}$ and radius 5.
::2. A圆圈,中心(6,4)和半径3;B圆圈,中心(-5,6)和半径5。

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3. Circle $\begin{align*}A\end{align*}$ with center $\begin{align*}(1, 4)\end{align*}$ and radius 2. Circle $\begin{align*}B\end{align*}$ with center $\begin{align*}(3, -2)\end{align*}$ and radius 7.
::3. A圆圈,中心(1,4)和半径2;B圆圈,中心(3,-2)和半径7。

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4. Circle $\begin{align*}A\end{align*}$ with center $\begin{align*}(8, 1)\end{align*}$ and radius 6. Circle $\begin{align*}B\end{align*}$ with center $\begin{align*}(6, -4)\end{align*}$ and radius 8.
::4. A圆圈,中心(8,1)和半径6.B圆圈,中心(6,4)和半径8。

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5. Circle $\begin{align*}A\end{align*}$ with center $\begin{align*}(-2, 10)\end{align*}$ and radius 3. Circle $\begin{align*}B\end{align*}$ with center $\begin{align*}(-1, -4)\end{align*}$ and radius 6.
::5. A圆圈,中心(-2,10)和半径3;B圆圈,中心(-1,-4)和半径6。

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6. Circle $\begin{align*}A\end{align*}$ with center $\begin{align*}(-1, 5)\end{align*}$ and radius 4. Circle $\begin{align*}B\end{align*}$ with center $\begin{align*}(-1, 5)\end{align*}$ and radius 5.
::6. A圆圈,中心(-1,5)和半径4.B圆圈,中心(-1,5)和半径5。

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7. Circle $\begin{align*}A\end{align*}$ with center $\begin{align*}(-4, -2)\end{align*}$ and radius 1. Circle $\begin{align*}B\end{align*}$ with center $\begin{align*}(1, 8)\end{align*}$ and radius 4.
::7. A圆圈,中心(-4,-2)和半径1;B圆圈,中心(1,8)和半径4。

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8. Circle $\begin{align*}A\end{align*}$ with center $\begin{align*}(10, 3)\end{align*}$ and radius 5. Circle $\begin{align*}B\end{align*}$ with center $\begin{align*}(4, 2)\end{align*}$ and radius 8.
::8. A圆圈,中心(10.3)和半径5;B圆圈,中心(4.2)和半径8。

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9. Circle $\begin{align*}A\end{align*}$ with center $\begin{align*}(12, 4)\end{align*}$ and radius 10. Circle $\begin{align*}B\end{align*}$ with center $\begin{align*}(12, 4)\end{align*}$ and radius 12.
::9. A圆圈,中心(12,4)和半径10;B圆圈,中心(12,4)和半径12。

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10. Circle $\begin{align*}A\end{align*}$ with center $\begin{align*}(-7, 6)\end{align*}$ and radius 9. Circle $\begin{align*}B\end{align*}$ with center $\begin{align*}(1, -4)\end{align*}$ and radius 9.
::10. A圆圈,中心(-7,6)和半径9;B圆圈,中心(1,-4)和半径9。

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11. The ratio of the circumference of circle $\begin{align*}A\end{align*}$ to the circumference of circle $\begin{align*}B\end{align*}$ is $\begin{align*}\frac{2}{3}\end{align*}$ . What is the ratio of their radii?
::11. A圈环绕与B圈环绕之比是23,其弧度之比是多少?

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12. The ratio of the area of circle $\begin{align*}A\end{align*}$ to the area of circle $\begin{align*}B\end{align*}$ is $\begin{align*}\frac{6}{1}\end{align*}$ . What is the ratio of their radii?
::12. A圆区域与B圆区域之比是61,其弧度之比是多少?

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13. The ratio of the radius of circle $\begin{align*}A\end{align*}$ to the radius of circle $\begin{align*}B\end{align*}$ is $\begin{align*}\frac{5}{9}\end{align*}$ . What is the ratio of their areas?
::13. 圆A半径与圆B半径之比是59,其区域之比是多少?

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14. The ratio of the area of circle $\begin{align*}A\end{align*}$ to the area of circle $\begin{align*}B\end{align*}$ is $\begin{align*}\frac{12}{5}\end{align*}$ . What is the ratio of their circumferences?
::14. A圆区域与B圆区域之比为125。 圆区域周长之比是多少?

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15. To show that any two circles are similar you need to perform a translation and/or a dilation. Why won't you ever need to use reflections or rotations?
::15. 要显示任何两个圆圈是相似的,你必须翻译和/或放大。为什么你从来不需要使用反射或旋转?

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16. In the diagram below, points $\begin{align*}A\end{align*}$ , $\begin{align*}B\end{align*}$ , $\begin{align*}C\end{align*}$ and $\begin{align*}D\end{align*}$ are each at the center of the circle. If the circles are all congruent, what is the relationship between the area of the square $\begin{align*}ABCD\end{align*}$ and the sum of the areas of all four circles?
::16. 在下图中,A、B、C和D点分别位于圆圈的中心,如果圆圈都一致,那么ABCD平方区域与所有四个圆圈区域之和之间的关系是什么?

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17. You have placed a safety light on the circumference of your bike tire. Describe the path of the light as you go down and then up a hill. Use words and a diagram.
::17. 你对自行车轮胎的环绕设置了安全灯,描述下坡然后上山的光线路径,使用文字和图表。

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Review (Answers)
::审查(答复)

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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 。