坐标校对

播放段落
• Prove or disprove that the quadrilateral defined by the points $\begin{align*}(2,4),(1,2),(5,1),(4,-1)\end{align*}$ is a .
  ::证明或否定各点界定的四边形(2,4,(1,2,2),(5,1),(4,-1)为a。
播放段落
• Prove or disprove that the point $\begin{align*}\left ( 1,\sqrt{7} \right )\end{align*}$ lies on the circle centered at the origin containing the point $\begin{align*}(0,4)\end{align*}$ .
  ::证明或反驳该点(1,7)位于原点(0,4)所在圆圆的圆上。
播放段落
• Prove or disprove that the quadrilateral defined by the points $\begin{align*}(8,-4),(0,2),(-10,2),(-6,4)\end{align*}$ is a trapezoid .
  ::证明或否定各点(8,-4,(0,2),(-10,2),(-6,4)所定义的四边形是。

播放段落

In order to be successful with coordinate proofs, you need to first remember the definitions and properties of different shapes. This is because you need to know what must be shown in order to prove or disprove the statement. For example, to prove that a quadrilateral is a parallelogram, you would need to show that the opposite sides are parallel .
::为了成功使用协调证明, 您需要首先记住不同形状的定义和属性。 这是因为您需要知道要证明或反驳该语句必须显示什么。 例如, 要证明四边形是平行图, 您需要显示对立面是平行的 。

播放段落

Next, you will need to figure out how you can use algebra to verify that a given shape has the necessary properties. The table below summarizes some of the common properties or facts you might want to show, and how to do it with algebra. Note that there is often more than one way to complete a coordinate proof!
::接下来, 您需要找出如何使用代数来验证某个形状是否具有必要的属性。 下表概述了您可能想要显示的一些共同属性或事实,以及如何用代数来进行。 请注意, 通常有不止一种方法来完成协调验证 !

播放段落 I want to show that... ::我想证明...	播放段落 With the help of algebra I should... ::在代数的帮助下,我应该...
播放段落 The shape has congruent sides. ::形状有相容的一面。	播放段落 Use the distance formula or Pythagorean Theorem to find the length of the sides and show that they are equal. ::使用距离公式或毕达哥里安定理词来找到边的长度并显示它们相等 。
播放段落 The shape has right angles . ::形状有正确的角度 。	播放段落 Show that the sides are perpendicular by finding the slopes of the lines (they should be opposite reciprocals). ::显示两侧垂直,找到线的斜坡(它们应该是对等的)。
播放段落 The shape has parallel sides. ::形状是平行的	播放段落 Show that the sides are parallel by finding the slopes of the lines (they should be equal). ::通过找到线的斜坡来显示两边的平行(它们应该是相等的) 。
播放段落 A point is on a shape. ::一个点在形状上。	播放段落 Find the equation(s) of the shape and verify that the point satisfies the equation. ::查找形状的方程并核实点是否满足方程。

播放段落

Let's do a few proofs together.
::让我们一起做一些证据。

播放段落

1. Prove or disprove that the quadrilateral defined by the points $\begin{align*}(2,4),(1,2),(5,1),(4,-1)\end{align*}$ is a parallelogram.
::1. 证明或否定各点界定的四边形(2,4,(1,2,2,2,5,1,4,1)是平行图。

播放段落

You don't necessarily have to plot the points, but it helps to visualize. It also allows you to check whether or not it looks like a parallelogram.
::您不必绘制点, 但它有助于视觉化。 它还允许您检查它是否像一个平行图。

播放段落

This definitely looks like a parallelogram. To prove that it is a parallelogram, remember that the definition of a parallelogram is a quadrilateral with two pairs of parallel sides. Therefore, one way to prove it is a parallelogram is to verify that the opposite sides are parallel. From algebra, remember that two lines are parallel if they have the same slope.
::这绝对看起来像一个平行图。 为了证明它是一个平行图, 请记住平行图的定义是带有两对平行面的四边形。 因此, 证明它的一个方法就是验证对立面是平行的。 从代数看, 请记住两条线是平行线, 如果两条线是相同的斜面的话 。

播放段落

Slope of $\begin{align*}\overline{AB}\end{align*}$ : $\begin{align*}m_1 = \frac{4-2}{2-1} = \frac{2}{1} = 2\end{align*}$
::面积: m1=4 - 22 - 1=21=2

播放段落

Slope of $\begin{align*}\overline{CD}\end{align*}$ : $\begin{align*}m_2 = \frac{-1-1}{4-5} =\frac{-2}{-1} = 2\end{align*}$
::面积: m2 #1 -14 -5 #2 -1=2

播放段落

Slope of $\begin{align*}\overline{BC}\end{align*}$ : $\begin{align*}m_3 = \frac{2-(-1)}{1-4} =\frac{3}{-3}= -1\end{align*}$
::bC: m3=2 -(-1)1 -4=3 -3 *1

播放段落

Slope of $\begin{align*}\overline{DA}\end{align*}$ : $\begin{align*}m_4 = \frac{1-4}{5-2} = \frac{-3}{3} = -1\end{align*}$
::面积 : m4 = 1 - 45 - 2 33 1

播放段落

The slopes of the opposite sides are equal. Therefore, the opposite sides are parallel . Therefore, the quadrilateral is a parallelogram .
::对面的斜坡是相等的。 因此对面的斜坡是平行的。 因此,四边线是平行的。

播放段落

2. Prove or disprove that the point $\begin{align*}\left (1,\sqrt{7} \right )\end{align*}$ lies on the circle centered at the origin containing the point $\begin{align*}(0,4)\end{align*}$ .
::2. 证明或否定该点(1,7)位于原点(0,4)。

播放段落

First, find the equation of the circle. The general equation of a circle is:
::首先,找到圆的方程。圆的一般方程是:

播放段落

$\begin{align*}(x-h)^2+(y-k)^2=r^2\end{align*}$
:x-h)2+(y-k)2=r2

播放段落

where $\begin{align*}(h,k)\end{align*}$ is the center of the circle and $\begin{align*}r\end{align*}$ is the radius of the circle. This circle is centered at the origin, so $\begin{align*}(h,k)=(0,0)\end{align*}$ . The circle contains the point $\begin{align*}(0,4)\end{align*}$ . From $\begin{align*}(0,0)\end{align*}$ to $\begin{align*}(0,4)\end{align*}$ is a distance of 4 units, so the radius of the circle must be 4. The equation of the circle is:
::此处 (h, k) 是圆的中心, r 是圆的半径。 此圆以原点为中心, 所以( h, k) =( 0, 0) 。 圆包含点 (0, 4) 。 从 0, 0 到 0, 4) 是 4 个单位的距离, 因此圆的半径必须是 4 。 圆的方程是 :

播放段落

$\begin{align*}x^2+y^2=16\end{align*}$
::x2+y2=16

播放段落

Now, to prove or disprove that the point $\begin{align*}\left (1,\sqrt{7} \right )\end{align*}$ lies on the circle, see if the point satisfies the equation of the circle:
::现在,要证明或反驳 点(1, 7) 位于圆圈上, 看看该点是否满足圆圈的等式 :

$\begin{align*}(1)^2+ \left ( \sqrt{7} \right )^2 &=16(?)\\ 1+7 &=16(?)\\ 8 &\ne 16 \end{align*}$

播放段落

The point does not satisfy the equation. Therefore, the point does NOT lie on the circle.
::点不能满足等式。 因此, 点不在圆上 。

播放段落

3. Prove or disprove that the quadrilateral defined by the points $\begin{align*}(8,1),(6,9),(4,0),(2,8)\end{align*}$ is a rectangle .
::3. 证明或否定各点界定的四边形(8,1,(6,9),(4,4),(2,8)是一个矩形。

播放段落

It helps to start by plotting the points.
::它有助于从绘制要点开始。

播放段落

This shape appears to be a rectangle. To prove that it is a rectangle, remember that the definition of a rectangle is a quadrilateral with four right angles. Therefore, to prove it is a rectangle you must verify that all angles are right angles. From algebra, remember that two lines meet at right angles if they are perpendicular, and two lines are perpendicular if they have opposite reciprocal slopes.
::此形状似乎是一个矩形。 要证明它是一个矩形, 请记住矩形的定义是一个四边形, 有四个右角度。 因此, 要证明它是一个矩形, 您必须验证所有角度都是正确的角度。 从代数 中, 请记住, 两条线如果是直角, 则在右角度相交, 如果两条线是对等斜度, 则直线是垂直的 。

播放段落

Slope of $\begin{align*}\overline{AB}\end{align*}$ : $\begin{align*}m_1 = \frac{1-9}{8-6} = \frac{-8}{2} = -4\end{align*}$
::面积: m1 = 1 - 98 - 6 82 * 4

播放段落

Slope of $\begin{align*}\overline{BC}\end{align*}$ : $\begin{align*}m_2 = \frac{9-8}{6-2} = \frac{1}{4}\end{align*}$
::bC: m2=9 - 86 - 2=14

播放段落

Slope of $\begin{align*}\overline{CD}\end{align*}$ : $\begin{align*}m_3 = \frac{8-0}{2-4} = \frac{8}{-2} = -4\end{align*}$
::面积: m3=8 - 02 - 4=8 - 2 *4

播放段落

Slope of $\begin{align*}\overline{DA}\end{align*}$ : $\begin{align*}m_4 = \frac{0-1}{4-8} = \frac{-1}{-4} = \frac{1}{4}\end{align*}$
::面积: m4=0 - 14 - 8 *1 - 4=14

播放段落

The slopes of each pair of adjacent sides are opposite reciprocals. Therefore, adjacent sides are perpendicular . Therefore, adjacent sides meet at right angles. Therefore, the quadrilateral is a rectangle .
::每对相邻两侧的斜坡是对的,因此,相邻两侧是垂直的,相邻两侧是右角的。因此,四边是矩形。

播放段落

Does the point $\begin{align*}P (-2.5, 3)\end{align*}$ lie inside, outside or on the circle $\begin{align*}x^2 + y^2 = 25\end{align*}$ ?
::P(- 2. 5, 3) 点是否位于 x2+y2=25 圆内、 外部或圆上?

播放段落

The point $\begin{align*}P (-2.5, 3)\end{align*}$ lies the circle.
::P(2.5,3)点为圆。

播放段落

Examples
::实例实例实例实例

播放段落

Example 1
::例1

播放段落

Sherry just bought some walkie talkies that have a one mile range. She rides her bike to go meet her friend Matt. If she gives Matt one of the walkie talkies, will she be able to talk to him on her walkie talkie from her house? Model this situation with an equation or inequality and a graph. Then, answer the question.
::雪莉刚买了一些有1英里距离的对讲机。 她骑着自行车去见她的朋友马特。 如果她给了马特一个对讲机,她能用她家的对讲机和他交谈吗? 用公式或不平等和图表来模拟这种情况。然后回答问题。

播放段落

If you imagine all the points within one mile of Sherry's house (in the range of the walkie talkies), you will have a circle . If you let Sherry's house be the origin, the set of points that Sherry could connect to with her walkie talkie is modeled by the inequality $\begin{align*}x^2+y^2\le 1\end{align*}$ .
::如果你想象雪利家一英里内的所有点数(在对讲机范围内), 你会有一个圆圈。 如果你让雪利家的家成为源头, 雪利可以与她的对讲机连接的点数会以不平等 x2+y2+1为模型。

播放段落

While Sherry is at home working on her biology project on bugs, Matt can move around with the walkie talkie and help her gather data.
::当雪莉在家研究有关虫子的生物项目时 马特可以和对讲机一起四处走动 帮她收集数据

播放段落

Click on the blue arrow and find out if Sherry is able to communicate with Matt using her walkie talkie? Drag the orange point to test if Matt's location is within the range of the walkie talkies.
::点击蓝色箭头, 并找出雪莉能否使用对讲机与马特沟通? 拖一下橙色点, 测试马特的位置是否在对讲机的范围之内 。

播放段落

So from the interactive above, you can verify that Matt is within the range graphically if the point lies within the shaded region. Also, you can verify this is within the range algebraically by testing to see if the point makes the inequality true.
::因此,从上面的交互关系中,你可以用图形来验证Matt是否在范围之内,如果该点位于阴影区域之内。此外,您也可以通过测试来验证该点是否使不平等成为真实,以此来验证它是否在范围内。

播放段落

You have now verified both algebraically and graphically that Matt's house is within the range of the walkie talkies.
::你现在已经用代数和图形 证实Matt的房子 在对讲机范围内

播放段落

Prove or disprove that the quadrilateral defined by the points $\begin{align*}(4,0)\end{align*}$ , $\begin{align*}(5,3)\end{align*}$ , $\begin{align*}(1,1)\end{align*}$ , $\begin{align*}(2,4)\end{align*}$ is a square .
::证明或否定各点(4,0,5,3,1,1,2,4)界定的四边形为正方形。

播放段落

First, plot the points. Below, the points have been labeled with letters to help with the identification of them later.
::首先,绘制点数。下面,点数贴上字母标签,以便稍后帮助确定点数。

播放段落

This shape appears to be a square. To prove that it is a square, remember that the definition of a square is a quadrilateral with four congruent sides and four right angles. Therefore, to prove it is a square you must:
::此形状似乎是一个正方形。 要证明它是一个正方形, 请记住一个正方的定义是一个四边形, 有四个正方形和四个右角度。 因此, 要证明它是一个正方形, 您必须:

播放段落

FIRST: Find the lengths of all sides and verify that they are equal. You can use the distance formula or to do this.
::第一: 查找所有边的长度, 并验证它们是否相等。 您可以使用距离公式或者这样做 。

播放段落

$\begin{align*}\overline{AB}=\sqrt{(5-2)^2 +(3-4)^2} = \sqrt{(3)^2+(-1)^2}=\sqrt{9 +1} = \sqrt{10}\end{align*}$
::===================================================================================================================

播放段落

$\begin{align*}\overline{BC}= \sqrt{(4-5)^2 +(0-3)^2} = \sqrt{(-1)^2 +(-3)^2}=\sqrt{1+9} = \sqrt{10}\end{align*}$
::=============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================

播放段落

$\begin{align*}\overline{CD}= \sqrt{(1-4)^2 +(1-0)^2} = \sqrt{(-3)^2+(1)^2}= \sqrt{9+1} = \sqrt{10}\end{align*}$
::=============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================

播放段落

$\begin{align*}\overline{DA}= \sqrt{(2-1)^2 + (4-1)^2} = \sqrt{(1)^2 +(3)^2}=\sqrt{1+9} = \sqrt{10}\end{align*}$
::==============================================================================================================================================

播放段落

Because all four sides are the same length, all four sides are congruent .
::因为四方的长度相同,四方的长度都一致。

播放段落

SECOND: Find the slopes of all four sides and verify that adjacent sides have opposite reciprocal slopes and therefore are perpendicular, creating right angles.
::第二,找到所有四面的斜坡,并核实相邻两侧的斜坡是对的,因此是垂直的,从而创造出正确的角度。

播放段落

Slope of $\begin{align*}\overline{AB}: m_1= \frac{3-4}{5-2} = -\frac{1}{3}\end{align*}$
::缩写 AB: m1 = 3 - 45 - 2 # 13

播放段落

Slope of $\begin{align*}\overline{BC}: m_2 = \frac{0-3}{4-5}= \frac{-3}{-1} = 3\end{align*}$
::BC: m2=0 - 34 - 5 - 5 - 3 - 1=3

播放段落

Slope of $\begin{align*}\overline{CD}: m_3 =\frac{1-0}{1-4} = -\frac{1}{3}\end{align*}$
::缩写 CD: m3 = 1 - 01 - 4 = 13

播放段落

Slope of $\begin{align*}\overline{DA}: m_4 = \frac{4-1}{2-1}=\frac{3}{1} =3\end{align*}$
::缩写 : m4=4 - 12 - 1=31=3

播放段落

Because all adjacent sides have opposite reciprocal slopes, adjacent sides are perpendicular . This means that adjacent sides meet at right angles and the shape has four right angles.
::由于所有相邻的两侧都有对等斜坡,相邻的两侧是垂直的,这意味着相邻的两侧在右角相交,形状有四个右角。

播放段落

The shape has four congruent sides and four right angles. Therefore, it is a square.
::形状有四个相似的侧面和四个右角度。 因此, 它是一个正方形 。

播放段落

Example 3
::例3

播放段落

Prove or disprove that the quadrilateral defined by the points $\begin{align*}(5,-1)\end{align*}$ , $\begin{align*}(6,3)\end{align*}$ , $\begin{align*}(1,1)\end{align*}$ , $\begin{align*}(2,5)\end{align*}$ is a rhombus .
::证明或否定各点(5,-1)界定的四边形(6,3)(1,1)(2,5)为暴风雪。

播放段落

First, plot the points.
::首先, 绘制点数 。

播放段落

This shape appears to be a rhombus. To prove that it is a rhombus, remember that the definition of a rhombus is a quadrilateral with four congruent sides. Therefore, to prove it is a rhombus you must verify that all sides are the same length. You can use the distance formula or the Pythagorean Theorem to do this.
::此形状似乎是一个曲折。 要证明它是一个曲折, 请记住, 曲折的定义是四边形的四边形。 因此, 要证明它是一个曲折, 您必须验证每边的长度都是一样的。 您可以使用距离公式或 Pythagorean 理论来做到这一点 。

播放段落

$\begin{align*}\overline{AB}=\sqrt{(6-2)^2 + (3-5)^2} = \sqrt{(4)^2+(-2)^2}=\sqrt{16+4} = \sqrt{20}= 2 \sqrt{5}\end{align*}$
::AB(6-2)2+(3-5)2+(3-5)2+(4)2+(-2)2+16+4=20=25

播放段落

$\begin{align*}\overline{BC}=\sqrt{(5-6)^2 + ((-1)-3)^2} = \sqrt{(-1)^2 + (-4)^2}=\sqrt{1+16} = \sqrt{17}\end{align*}$
::==BC_(5 - 6)2+((1 - 1 - 3)2 _( 1)2+( - 4)2+1+( - 4)2+1+16+_17

播放段落

$\begin{align*}\overline{CD}=\sqrt{(1-5)^2 + (1-(-1))^2} = \sqrt{(-4)^2+(2)^2}=\sqrt{16+4} = \sqrt{20} = 2 \sqrt{5}\end{align*}$
::======================================================================================================================================================================================================================================================================

播放段落

$\begin{align*}\overline{DA}=\sqrt{(2-1)^2+(5-1)^2} = \sqrt{(1)^2 + (4)^2} = \sqrt{1+16} = \sqrt{17}\end{align*}$
::======================================================================================================================================================

播放段落

Even though the shape looked like a rhombus, its four sides are not congruent . Therefore, this is NOT a rhombus.
::尽管形状看起来像暴风雨,但四面并不一致。 因此,这不是暴风雨。

播放段落

Example 4
::例4

播放段落

Prove or disprove that the quadrilateral defined by the points $\begin{align*}(8,-4)\end{align*}$ , $\begin{align*}(0,2)\end{align*}$ , $\begin{align*}(-10,2)\end{align*}$ , $\begin{align*}(-6,4)\end{align*}$ is a trapezoid.
::证明或否定各点(8,-4)界定的四边形(0,2),(-10,2),(-6,4)是。

播放段落

It helps to start by plotting the points.
::它有助于从绘制要点开始。

播放段落

This shape appears to be a trapezoid. To prove that it is a trapezoid, remember that the definition of a trapezoid is a quadrilateral with exactly one pair of parallel sides. Therefore, to prove it is a trapezoid you must verify that one pair of sides is parallel. From algebra, remember that two lines are parallel if they have the same slope.
::此形状似乎是一个陷阱类体。 要证明它是一个陷阱类体, 请记住, 陷阱类体的定义是一个四边形, 具有一对齐的平行边。 因此, 要证明它是一个陷阱类体, 您必须验证一对齐。 从代数来看, 请记住, 两条线是平行的, 如果两条线是相同的斜度 。

播放段落

Slope of $\begin{align*}\overline{AB}\end{align*}$ : $\begin{align*}m_1 = \frac{2-4}{0-(-6)} = \frac{-2}{6} = -\frac{1}{3}\end{align*}$
::缩写 AB: m1=2 -40 -(-6)\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ AB: m1=2 -40\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\13

播放段落

Slope of $\begin{align*}\overline{BC}\end{align*}$ : $\begin{align*}m_2 = \frac{-4-2}{8-0} = \frac{-6}{8} = -\frac{3}{4}\end{align*}$
::缩写 : m24 - 28 - 068 - 34

播放段落

Slope of $\begin{align*}\overline{CD}\end{align*}$ : $\begin{align*}m_3 = \frac{2-(-4)}{-10-8} = \frac{6}{-18} = - \frac{1}{3}\end{align*}$
::m3=2 -( - 4) - - 10 - 8=6 - 18=13

播放段落

Slope of $\begin{align*}\overline{DA}\end{align*}$ : $\begin{align*}m_4 = \frac{4-2}{-6-(-10)} = \frac{2}{4} = \frac{1}{2}\end{align*}$
::面积 : m4=4 - 2 - 6 - ( - 10)=24=12

播放段落

The slopes of exactly one pair of sides are equal. Therefore, exactly one pair of opposite sides is parallel. Therefore, the quadrilateral is a trapezoid.
::一对两面的斜坡是相等的。 因此,一对对对对的斜坡是平行的。 因此,四边形是一个陷阱。

播放段落

CK-12 PLIX Interactive
::CK-12 PLIX 交互式互动

Summary

播放段落 In a coordinate proof , you are proving geometric statements using algebra and the coordinate plane. ::在坐标证明中,你正在用代数和坐标平面来证明几何语句。 播放段落 Some common proofs involve the following definitions: 播放段落 Congruent figures are identical in size, shape and measure. Shapes can still be congruent if they are rotated. ::相容的数字大小、形状和量度相同。 形状如果旋转, 仍然可以相同 。 播放段落 A quadrilateral is a closed figure with four sides and four vertices. ::一个四边形是一个有四面和四个顶点的封闭数字。 播放段落 A parallelogram is a quadrilateral with two pairs of parallel sides. A parallelogram may be a rectangle, a rhombus, or a square. ::平行图是两对平行面的四边形。 平行图可以是矩形、 圆柱形或方形。 ::一些常见证据涉及以下定义: Congruent 数字在大小、形状和度量方面是相同的。 形状在旋转时仍然可以相同。 四方是四面和四个顶点的封闭形。 平行图是两对平行面的四边形。 平行图可以是矩形、 圆柱形或方形。

播放段落

1. Prove or disprove that the quadrilateral defined by the points $\begin{align*}(5,3),(3,5),(3,1),(1,3)\end{align*}$ is a square.
::1. 证明或否定各点(5,3,3(3,5),(3,1,1,(1,3))界定的四边形为方形。

播放段落

2. Prove or disprove that the quadrilateral defined by the points $\begin{align*}(7,6),(6,8),(2,3),(1,5)\end{align*}$ is a rectangle.
::2. 证明或否定各点(7,6,(6),(8),(2),(3),(1,5)界定的四边形是一个矩形。

播放段落

3. Prove or disprove that the quadrilateral defined by the points $\begin{align*}(4,0),(0,3),(0,-3),(-4,0)\end{align*}$ is a rhombus.
::3. 证明或否定各点(4,0,(0),(0),(3),(4),(4,0)所定义的四边形是暴风雨。

播放段落

4. Prove or disprove that the quadrilateral defined by the points $\begin{align*}(6,4),(1,3),(3,2),(-2,1)\end{align*}$ is a parallelogram.
::4. 证明或否定各点界定的四边形(6,4,(1,3,3,3,2,2,1)是平行图。

播放段落

5. Prove or disprove that the quadrilateral defined by the points $\begin{align*}(2,1),(0,4),(-6,-2),(-4,3) \end{align*}$ is a trapezoid.
::5. 证明或否定各点界定的四边形(2,1,(0,4),(-6),(-2),(-4),(3)是。

播放段落

6. Prove or disprove that the quadrilateral defined by the points $\begin{align*}(5,3),(5,8),(2,2),(1,5)\end{align*}$ is a kite.
::6. 证明或否定各点(5、3、5、8、2、2、1、5)界定的四边形是风筝。

播放段落

7. Prove or disprove that the quadrilateral defined by the points $\begin{align*}(0,-2),(-1,1),(-3,-3),(-4,0)\end{align*}$ is a square.
::7. 证明或否定各点(0,-2,(-1,1),(-3,3),(-4,0)界定的四边形为正方形。

播放段落

8. Prove or disprove that the quadrilateral defined by the points $\begin{align*}(7,2),(5,4),(4,1),(2,3) \end{align*}$ is a rhombus.
::8. 证明或否定各点(7,2,(5,4),(4),(1),(2,3)界定的四边形是暴风雪。

播放段落

9. Prove or disprove that the quadrilateral defined by the points $\begin{align*}(5,-1),(6,3),(1,1),(2,5) \end{align*}$ is a parallelogram.
::9. 证明或否定由点(5,-1,(6),(3),(1,1),(2,5)界定的四边形是一个平行图。

播放段落

10. Prove or disprove that the quadrilateral defined by the points $\begin{align*}(6,2),(3,4),(4,-1),(1,1) \end{align*}$ is a rectangle.
::10. 证明或否定各点界定的四边形(6,2,(3,4),(4)-1,(1,1)是一个矩形。

播放段落

11. Prove or disprove that the point $\begin{align*}(2,4)\end{align*}$ lies on the exterior of the circle centered at the point $\begin{align*}(1,2)\end{align*}$ that passes through the point $\begin{align*}(3,3)\end{align*}$ .
::11. 证明或反驳,点(2,4)位于圆的外部,圆的中心(1,2)穿过点(3,3)。

播放段落

12. Prove or disprove that the point $\begin{align*}(-6,2)\end{align*}$ lies on the circle centered at the point $\begin{align*}(-3,-1)\end{align*}$ that passes through the point $\begin{align*}(0,2)\end{align*}$ .
::12. 证明或否定该点(-6.2)位于通过点(0.2)的圆(-3-1)的中心圆上。

播放段落

13. Prove or disprove that the point $\begin{align*}(4,-1)\end{align*}$ lies on the interior of the circle centered at the point $\begin{align*}(3,2)\end{align*}$ that passes through the point $\begin{align*}(1,-1)\end{align*}$ .
::13. 证明或反驳,该点(4,-1)位于圆的内部,圆的中心点(3,2)穿过点(1,-1),或圆的内部。

播放段落

14. Prove or disprove that the point $\begin{align*}\left ( 1,2\sqrt{2}\right )\end{align*}$ lies on the circle centered at the origin that passes through the point $\begin{align*}(0,3)\end{align*}$ .
::14. 证明或反驳,该点(1,22)位于以通过点(0,3)的起源为中心的圆圈上。

播放段落

15. Prove or disprove that the point $\begin{align*}(1,7.5)\end{align*}$ lies on the interior of the circle centered at the point $\begin{align*}(1,6)\end{align*}$ that passes through the point $\begin{align*}(2,5)\end{align*}$ .
::15. 证明或反驳,该点(1,7.5)位于圆心(1,6)穿过点(2,5)的圆心的内部。

播放段落

16. AB and CD are two chords of a circle with center O. Knowing that $\begin{align*}\overline{AB}\bot \overline{CD}\end{align*}$ , prove that $\begin{align*}\angle AOD\end{align*}$ and $\begin{align*}\angle BOC\end{align*}$ are supplementary.
::AB和CD是O中心圆圈的两个和弦。 知道'AB'CD, 证明'AOD'和'BOC'是补充的。

播放段落

17. Prove that a given parallelogram inscribed in a circle is a rectangle.
::17. 证明圆内刻有的平行图是一个矩形。

播放段落

18. Given the point A( t , 3 t - 12), prove or disprove that A is equidistant from B(0, 3) and C(9, 0) for all values of t . Describe the figure formed by the points A for all values of t . What is the relationship between this figure and the points B and C?
::18. 鉴于A(t, 3t - 12)点,证明或证明A与B(0)和C(9,0)的所有数值的距离相等。 描述A点对t的所有数值构成的数字。 这个数字与B和C点之间的关系是什么?

播放段落

Review (Answers)
::审查(答复)

播放段落

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 。