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  • 坐标平面上的多边形分类

    lesson content

    Parker is taking her first drawing class, though she doesn't have high hopes. When she walks into class the first day, her art teacher pulls up a picture of a cityscape and tells them that they will all be able to draw this by the end of the week. Her teacher gives them each a copy of this picture and tells them that their first assignment is to break the picture into its underlying shapes. Parker is utterly confused and overwhelmed. How can Parker figure out how to break the picture into shapes?
    ::Parker正在上第一个绘画班,虽然她没有很高的希望。当她第一天上课时,她的艺术老师拿出了一幅城市景象的照片,告诉他们,他们都能在周末画出。她的教师给他们每人一张图片,告诉他们,他们的第一个任务是把图片打碎成其基本形状。Parker完全困惑和不堪重负。Parker如何找到如何把图片打成形状?

    In this concept, you will learn how to graph geometric figures given coordinates of  vertices  and identify graphed figures.
    ::在此概念中,您将学会如何绘制给定脊椎坐标的几何数字,并确定图表数字。

    Classifying Polygons in the Coordinate Plane
    ::坐标平面中对多边形进行分类

    A  coordinate grid  is a grid in which points are graphed. It usually has two or more  intersecting lines  which divide a  plane  into quadrants, and in which ordered pairs, or coordinates, are defined. It usually has four  quadrants,  or sections, to it. 
    ::坐标网格是一个用图形绘制点数的网格。 它通常有两个或两个以上的交叉线, 将平面分为四重方, 并定义定购的配对或座标。 它通常有四个四重方, 或四分段 。

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    The  origin  is the place where the two lines intersect. Its coordinates are defined as (0,0).
    ::来源是两条线交叉的地方。其坐标被定义为(0,0) 。

    The  x-axis  is the  line  running from left to right that has the numbers defined on it and is usually labeled with an "x". The x-coordinate of an ordered pair is found with relation to it. All the points located on the x-axis have a y-coordinate of 0.
    ::X 轴是从左向右运行的线条,其编号是定义的,通常贴有“x”的标签。与该线条相关的是定购一对的X坐标。X 轴上的所有点都有0的Y坐标。

    The  y-axis  is the central line that runs up-down and is labeled with a "y". Y-coordinates are plotted in reference to this axis. Again, all the x-coordinates of points located on the y-axis are 0.
    ::Y 轴是自上而下并贴有“y”标签的中心线。 Y 坐标是在此轴上绘制的。同样,Y 轴上的所有点的 X 坐标都是 0 。

    An  ordered pair  is a list of two numbers in parenthesis, separated by a comma like this: (5,-3). It tells where a  point  is located on the coordinate plane. The first number is the x-coordinate. It tells you where to go on the x-axis. If it is positive, you go to the right. If it is negative, you go to the left. The second number is the y-coordinate. It tells you where to go on the y-axis. If it is positive, you go up. If it is negative, you go down.
    ::订购的一对是括号中两个数字的列表,用像这样的逗号分隔开来: (5, 3) 。 它说明坐标平面上的点的位置。 第一个数字是 x 坐标。 它告诉您如何使用 x 轴。 如果是正数, 则向右走。 如果为负数, 则向左走。 第二个数字是 Y 坐标。 它告诉您如何使用 Y 轴。 如果为正数, 则向上。 如果为负数, 则向下走 。

    The   vertex  of a shape is the place where two sides of the shape come together. In general, when a shape is defined inside of a coordinate plane, it is defined by the vertices, and then the lines are drawn to connect them.
    ::形状的顶点是形状两面聚集的地方。 一般来说, 当一个形状在坐标平面内被定义时, 它由顶点来定义, 然后绘制线条来连接它们 。

    A  polygon  is any shape made up of rectilineal, or straight, lines. The smallest polygon is a  triangle , which has three sides. A five-sided figure is a  pentagon . And many polygons with more sides than five are also named.
    ::多边形是由直线或直线组成的任何形状。最小的多边形是一个三角形,有三面。五面图是五面形。许多边大于五面的多边形也被命名。

    A  right angle  is an  angle  that looks like where the axes on the coordinate plane meet.
    ::右角度是看似坐标平面上的轴相交角的角。

    Parallel lines  are lines that will go on forever but will never converge.
    ::平行的线条将永远延续,但永远无法汇合。

    The category of four-sided polygons includes:
    ::四面多边形类别包括:

    • the  square , which has four sides of equal length and its angles are all right angles;
      ::方形,其四面长度相等,其角度均为正确角度;
    • the  rhombus  which, like the square, has four equal sides, but is "tilty";
      ::象广场一样,有四个平等面的暴风雨,但是是“肥料”;
    • the  rectangle  which has two pairs of sides which are equal and all its angles are right angles;
      ::矩形,有两对相等的两边,所有角度都是正确的角度;
    • the  trapezoid  which may have no equal sides, but it has two lines which are  parallel ; and
      ::甲状腺素,可能没有相等的两边,但有两条平行线;和
    • the   which has two sets of parallel lines which are equal in length to each other (but like the rhombus, it is "tilty"). 
      ::具有两组平行线,彼此长度相等(但和暴风雨一样,它是“薄荷”)。

    In order to graph a figure in the coordinate plane, you just graph each of the vertices and then connect them with straight lines so that none of the lines cross. The number of sides you have is the same as the number of vertices. So a triangle, for example, is defined with three vertices.
    ::要绘制坐标平面上的图, 您只需绘制每个顶点的图, 然后用直线将其连接, 这样没有一条线横过。 您的边数与顶点数相同。 例如, 三角形定义为三个顶点 。

    Here is an example.
    ::举一个例子。

    Graph a figure with the coordinates  A ( − 4 , 3 )   B ( 2 , 3 )   C ( 2 , − 1 )   D ( − 4 , − 1 ) . When finished, name the figure that has been drawn on the grid.
    ::a 图中坐标A (-4, 3) B (2, 3) C (2, 1) D (-4, - 1) 的坐标A (-4, 3) B (2, 3) C (2, 1) D (-4, -1) 的坐标图。

    First, plot each point on the coordinate grid and then connect the lines.
    ::首先,在坐标网格上绘制每个点,然后连接线条。

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    Next, in order to determine what kind of shape it is, first count the number of vertices.
    ::接下来,为了确定它是什么形状, 首先计算顶点数 。

    This figure has four, so it is one of the four-sided shapes.
    ::这个数字有四个,所以是四面形的形状之一。

    Then, look to see how many of the sides are equal.
    ::那么,看看有多少 双方是平等的。

    In this case, side AB=DC and AD=BC. Since there are two sets of equal sides, this is either a rectangle or a parallelogram.
    ::在这种情况下,AB=DC和AD=BC。由于有两组平等方,这要么是一个矩形,要么是一个平行图。

    Finally, check the angles.
    ::最后,检查角度。

    In this shape, the angles are right angles, so this is a rectangle.
    ::在这个形状中,角度是右边的角度, 所以这是一个矩形 。

    Here is another example.
    ::下面是另一个例子。

    Graph and name the following figure with these coordinates  D ( 1 , 3 )   E ( 5 , 3 )   F ( 7 , − 1 )   G ( 1 , − 1 ) .
    ::下图并标明坐标D(1、3)E(5、3)F(7、1)G(1、1)G(1、-1)的下图。

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    First, plot the vertices and connect them.
    ::首先,绘制顶峰并连接它们。

    Next, count the number of vertices. 
    ::下一步,计数顶点数。

    There are four, so this is a four-sided figure, which you can see from its shape.
    ::有四个,所以这是一个四面形的图, 你可以从它的形状中看出来。

    Then, look at the sides.
    ::然后,看看两边。

    None of the sides are the same length, so this shape must be a trapezoid.
    ::每一边的长度都不相同, 所以这个形状必须是状的。

    Examples
    ::实例

    Example 1
    ::例1

    Earlier, you were given a problem about Parker and her math-art panic attack.
    ::早些时候,你得到一个问题 帕克和她数学专家的恐慌症发作。

    Her art teacher gives her a picture of buildings in perspective and tells her to break it into its constituent shapes.
    ::她的艺术老师给了她一幅建筑物的图画, 让她打破建筑的形状。

    First, she draws a y-axis down the  center  of the back building in the picture, which looks about halfway across the picture. Then, she draws the x-axis as the horizon line.
    ::首先,她画了一幅Y轴的画, 画中后面的建筑中间, 画中看中了一半。 然后,她画了X轴作为地平线。

    Next, she squints at the picture and tries to see it not as buildings, but as shapes, instead. On the right side of the picture, she realizes that the buildings make a triangle with one vertex starting at the origin. And on the left of the y-axis, the other side of buildings make another triangle, also with a vertex at the origin. Even the street, she realizes, makes its own triangle, again with one vertex at the origin. And then the sky finishes it with a forth triangle with a vertex at the origin.
    ::接着,她在图片上斜视,试图把它看成不是建筑,而是形状。在图片的右侧,她意识到这些建筑造了一个三角形,有一个顶点从原点开始。在Y轴的左侧,建筑的另一侧造另一个三角形,也有一个顶点。即使街道,她也意识到,它也造了自己的三角形,在原点有一个顶点。然后天空用一个前端三角形和原点的一个顶点来结束它。

    Then, she sees that the rest of the picture is just made up of  . There is one below the street triangle, and one on the left buildings. And one where the tiny back building is.
    ::然后,她看到照片的其余部分只是由...

    Parker concludes that she might be able to learn to draw after all.
    ::Parker认为她也许能学会画画

    Example 2
    ::例2

    Determine the shape given by the following vertices.
    ::确定以下顶点给定的形状。

    ( − 4 , 6 ) ( 4 , 6 ) ( 0 , − 6 )

    First, count the number of vertices.
    ::首先,计数顶点数。

    In this case, there are three vertices. If there are three vertices, then the shape has three sides.
    ::在这种情况下,有三个顶点。如果有三个顶点,那么形状是三面的。

    Next, determine which shapes are possible.
    ::下一步,决定哪些形状是可能的。

    There is only one closed shape with three sides, and that is a triangle.
    ::只有一个有三面的封闭形状,这是一个三角形。

    The answer is a triangle.
    ::答案是一个三角形。

    Example 3
    ::例3

    Determine the shape given by the following vertices.
    ::确定以下顶点给定的形状。

    ( 0 , 2 ) ( 2 , 0 ) ( 0 , − 3 )

    First, count the number of vertices.
    ::首先,计数顶点数。

    In this case, there are three vertices. If there are three vertices, then there are three sides.
    ::在这种情况下,有三个顶点。如果有三个顶点,则有三面。

    Next, determine which shapes are possible.
    ::下一步,决定哪些形状是可能的。

    There is only one closed shape with three sides, and that is a triangle.
    ::只有一个有三面的封闭形状,这是一个三角形。

    The answer is a triangle.
    ::答案是一个三角形。

    Example 4
    ::例4

    Determine the shape given by the following vertices.
    ::确定以下顶点给定的形状。

    ( − 1 , − 3 ) ( − 5 , − 3 ) ( − 7 , 1 ) ( − 1 , 1 )

    First, count the number of vertices.
    ::首先,计数顶点数。

    In this case, there are four vertices. If there are four vertices, then the shape has four sides.
    ::在这种情况下,有四个顶点。如果有四个顶点,则形状有四面。

    Next, determine which shapes are possible.
    ::下一步,决定哪些形状是可能的。

    The four-sided figures are: square, rhombus, trapezoid, rectangle, and parallelogram.
    ::四面数字为:正方形、正方形、正形、形、矩形和平行图。

    Then, plot the vertices on a coordinate plane.
    ::然后,在坐标飞机上绘制顶峰图。

    Then, determine the properties of the shape.
    ::然后确定形状的属性。

    In this case, the shape has two parallel lines and two non-parralel lines. The only four sides figure in which that is the case is the trapezoid.
    ::在这种情况下,形状有两条平行线和两条非平行线,只有四个侧面的图象是陷阱。

    The answer is a trapezoid.
    ::答案是一个陷阱状的。

    Example 5
    ::例5

    Determine the shape given by the following vertices.
    ::确定以下顶点给定的形状。

    ( 3 , 3 ) ( 0 , 3 ) ( 0 , 0 ) ( 3 , 0 )

    First, count the number of vertices.
    ::首先,计数顶点数。

    In this case, there are four vertices. If there are four vertices, then the shape has four sides.
    ::在这种情况下,有四个顶点。如果有四个顶点,则形状有四面。

    Next, determine which shapes are possible.
    ::下一步,决定哪些形状是可能的。

    The four-sided figures are: square, rhombus, trapezoid, rectangle, and parallelogram.
    ::四面数字为:正方形、正方形、正形、形、矩形和平行图。

    Then, plot the vertices on a coordinate plane.
    ::然后,在坐标飞机上绘制顶峰图。

    Then, determine the properties of the shape.
    ::然后确定形状的属性。

    In this case, the shape has two sets of parallel lines and and all the sides are the same length. Additionally, the shape has four right angles. The square is the shape that meets these criterion.
    ::在这种情况下,形状有两组平行线,两侧长度相同。此外,形状有四个右角度。方形是符合这些标准的形状。

    The answer is a square.

    ::答案是方形的。

    Review
    ::回顾

    Graph each figure using the vertices. Then name the graphed figure.
    ::使用顶点绘制每个图,然后标出图表。

    1. A ( − 2 , 2 ) ; B ( 2 , 2 ) ; C ( 2 , − 2 ) ; D ( − 2 , − 2 )  
      :sadA - 2, 2) ; B (2, 2) ; C (2, 2) ; C (2, 2) ; D (-2, - 2) ;
    2. D ( − 4 , 3 ) ; E ( − 1 , 1 ) ; F ( − 4 , 1 )
      :sadD - 4, 3 ) ; E ( - 1, 1 ) ; F ( - 4, 1 )
    3. ( 1 , 1 ) ( 2 , 3 ) ( 4 , 1 )
    4. ( − 1 , 3 ) ( − 5 , 3 ) ( − 1 , 0 ) ( − 5 , 0 )
    5. ( 0 , 5 ) ( 3 , 5 ) ( 0 , 9 ) ( 3 , 9 )
    6. ( 0 , 6 ) ( 2 , 6 ) ( 0 , 10 ) ( 2 , 10 )
    7. ( − 3 , 6 ) ( 6 , 0 ) ( 9 , 0 )
    8. ( − 1 , 6 ) ( − 1 , 8 ) ( − 9 , 6 ) ( − 9 , 8 )
    9. ( 0 , − 8 ) ( − 1 , − 5 ) ( 5 , − 5 ) ( 4 , − 8 )
    10. ( 12 , 0 ) ( 12 , 6 ) ( 7 , 0 )

    For 11 - 15 draw five of your own figures on a coordinate grid. Write out each set of coordinates and work with a partner to identify each figure using only the coordinates.
    ::对于 11 - 15 , 请在 坐标网 上 抽取 5 个您自己的 数字 。 请写下每一组坐标 , 并与 伙伴 合作 , 只用 坐标 来 确定 每一 个 数字 。