Drawing Polygons on the Coordinate Plane | 在坐标平面上绘制多边形

📚 Drawing Polygons on the Coordinate Plane | 在坐标平面上绘制多边形

In this lesson, we will explore how to draw polygons on the coordinate plane using given coordinates of their vertices. Understanding this skill is essential for solving geometry problems and visualising shapes in mathematics.

在本节课中，我们将学习如何使用给定的顶点坐标在坐标平面上绘制多边形。掌握这一技能对于解决几何问题和在数学中直观理解图形至关重要。

1. The Coordinate Plane Review | 坐标平面复习

The coordinate plane is formed by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). Their intersection is called the origin, which has coordinates (0, 0). Each point on the plane is identified by an ordered pair (x, y), where x represents the horizontal position and y represents the vertical position.

坐标平面由两条互相垂直的数轴组成：x轴（水平方向）和y轴（垂直方向）。它们的交点称为原点，坐标为(0, 0)。平面上的每个点都用有序数对(x, y)来表示，其中x表示水平位置，y表示垂直位置。

When plotting a point, always start at the origin. Move right for positive x values and left for negative x values. Then move up for positive y values and down for negative y values. For example, to plot the point (3, 2), move 3 units to the right and 2 units up.

在描点时，始终从原点开始。x值为正则向右移动，x值为负则向左移动。然后y值为正向上移动，y值为负向下移动。例如，要描点(3, 2)，向右移动3个单位，再向上移动2个单位。

2. What is a Polygon? | 什么是多边形？

A polygon is a closed two-dimensional shape made up of straight line segments connected end to end. Each segment is called a side, and each endpoint where two sides meet is called a vertex (plural: vertices). Common polygons include triangles (3 sides), quadrilaterals (4 sides), pentagons (5 sides), and hexagons (6 sides).

多边形是由首尾相连的直线段构成的封闭二维图形。每条线段称为边，两条边相交的端点称为顶点。常见的多边形包括三角形（3条边）、四边形（4条边）、五边形（5条边）和六边形（6条边）。

To draw a polygon on the coordinate plane, we plot all its vertices as points, then connect them in order with straight line segments. The last vertex must connect back to the first vertex to close the shape. It is important to connect the points in the order given, not in any arbitrary order.

要在坐标平面上绘制多边形，我们先将所有顶点作为点描出来，然后按顺序用直线段连接它们。最后一个顶点必须连接回第一个顶点以封闭图形。重要的是按照给定的顺序连接点，而不是任意连接。

3. Plotting Points Accurately | 准确描点

Accuracy is crucial when plotting points on the coordinate plane. A common mistake is reversing the x and y coordinates. Remember: x comes first, then y — always “along the corridor, then up the stairs”.

在坐标平面上描点时，准确性至关重要。一个常见的错误是颠倒x和y坐标。记住：x在前，y在后——始终“先沿走廊走，再上楼梯”。

Consider the points A(1, 2), B(4, 2), C(4, 5), and D(1, 5). To plot A(1, 2), we move 1 unit right along the x-axis, then 2 units up. To plot B(4, 2), we move 4 units right, then 2 units up. Each point must be plotted carefully with a small dot and labelled with its letter.

考虑点A(1, 2)、B(4, 2)、C(4, 5)和D(1, 5)。要描出A(1, 2)，沿x轴向右移动1个单位，再向上移动2个单位。要描出B(4, 2)，向右移动4个单位，再向上移动2个单位。每个点都必须仔细描出并用字母标注。

4. Drawing a Rectangle | 绘制长方形

Let us draw a rectangle using the points A(1, 2), B(4, 2), C(4, 5), and D(1, 5) from our previous example. After plotting all four points, connect A to B, B to C, C to D, and finally D back to A.

让我们用上一示例中的点A(1, 2)、B(4, 2)、C(4, 5)和D(1, 5)来绘制一个长方形。描出所有四个点后，连接A到B，B到C，C到D，最后D回到A。

Notice that AB is a horizontal segment of length 3 units (from x = 1 to x = 4), and AD is a vertical segment of length 3 units (from y = 2 to y = 5). Since opposite sides of a rectangle are equal and all angles are right angles, we can verify that our shape is indeed a square — a special type of rectangle where all sides are equal.

注意AB是长度为3个单位的水平线段（从x=1到x=4），AD是长度为3个单位的垂直线段（从y=2到y=5）。由于矩形对边相等且所有角均为直角，我们可以验证该图形实际上是一个正方形——一种所有边都相等的特殊矩形。

5. Drawing a Triangle | 绘制三角形

Triangles are the simplest polygons with three sides. Suppose we want to draw triangle PQR with vertices P(-3, 1), Q(2, 4), and R(2, -1). Plot each point, then connect P to Q, Q to R, and R to P.

三角形是只有三条边的最简单多边形。假设我们要绘制三角形PQR，顶点为P(-3, 1)、Q(2, 4)和R(2, -1)。描出每个点后，连接P到Q，Q到R，R到P。

When examining this triangle, we can observe that QR is a vertical segment because both Q and R share the same x-coordinate of 2. The length of QR is the difference in y-coordinates: 4 – (-1) = 5 units. To find the area of the triangle, we would need to identify the base and height — in this case, QR can serve as the base, and the height is the horizontal distance from P to the line x = 2, which is 2 – (-3) = 5 units.

观察这个三角形，我们可以发现QR是垂直线段，因为Q和R的x坐标都为2。QR的长度是y坐标之差：4 – (-1) = 5个单位。要求三角形的面积，需要确定底和高——在这种情况下，QR可以作为底边，高是P到直线x=2的水平距离，即2 – (-3) = 5个单位。

Area = ½ × base × height = ½ × 5 × 5 = 12.5 square units

面积 = ½ × 底 × 高 = ½ × 5 × 5 = 12.5 平方单位

6. Recognising Horizontal and Vertical Sides | 识别水平边和垂直边

When two points share the same y-coordinate, the line segment connecting them is horizontal. The length of this segment is the absolute difference of their x-coordinates. For example, the distance between A(2, 3) and B(7, 3) is |7 – 2| = 5 units.

当两个点具有相同的y坐标时，连接它们的线段是水平的。该线段的长度是它们x坐标之差的绝对值。例如，A(2, 3)和B(7, 3)之间的距离是|7 – 2| = 5个单位。

Similarly, when two points share the same x-coordinate, the line segment connecting them is vertical. The length is the absolute difference of their y-coordinates. For instance, the distance between C(4, 1) and D(4, 6) is |6 – 1| = 5 units.

类似地，当两个点具有相同的x坐标时，连接它们的线段是垂直的。长度是它们y坐标之差的绝对值。例如，C(4, 1)和D(4, 6)之间的距离是|6 – 1| = 5个单位。

These properties help us calculate side lengths and perimeters of polygons quickly without plotting the points, as long as the sides are horizontal or vertical.

这些性质帮助我们在不需要描点的情况下快速计算多边形的边长和周长，只要边是水平或垂直的即可。

7. Drawing a Quadrilateral with No Parallel Sides | 绘制没有平行边的四边形

Not all polygons have horizontal or vertical sides. Consider a quadrilateral with vertices W(-2, 1), X(1, 4), Y(4, 1), and Z(1, -2). Connect W to X, X to Y, Y to Z, and Z to W.

并非所有多边形都有水平或垂直的边。考虑一个四边形，顶点为W(-2, 1)、X(1, 4)、Y(4, 1)和Z(1, -2)。连接W到X，X到Y，Y到Z，Z到W。

This shape is a quadrilateral known as a kite. Notice that WX = XY and YZ = ZW. The diagonals WY and XZ are perpendicular to each other. This example shows that even without horizontal or vertical sides, we can still identify properties of the shape by examining the coordinates.

这个图形是一个称为风筝形的四边形。注意WX = XY，YZ = ZW。对角线WY和XZ互相垂直。这个例子表明，即使没有水平或垂直的边，我们仍然可以通过检查坐标来识别图形的性质。

8. Perimeter and Area on the Coordinate Plane | 坐标平面上的周长与面积

Once a polygon is drawn, we can calculate its perimeter by adding the lengths of all sides. For horizontal and vertical sides, use coordinate differences as shown earlier. For slanted sides, we will learn in later lessons how to use the Pythagorean theorem — for now, we focus on polygons with horizontal and vertical sides only.

画好多边形后，我们可以通过将所有边的长度相加来计算周长。对于水平和垂直的边，使用前面展示的坐标差来计算。对于斜边，我们将在后续课程中学习如何使用勾股定理——目前先聚焦于只有水平边和垂直边的多边形。

To find the area of a rectangle or square on the coordinate plane, multiply its length by its width. For example, a rectangle with vertices at (0, 0), (4, 0), (4, 3), and (0, 3) has length 4 units and width 3 units, so its area is 4 × 3 = 12 square units.

要计算坐标平面上长方形或正方形的面积，将其长乘以宽。例如，一个顶点为(0, 0)、(4, 0)、(4, 3)和(0, 3)的长方形，长为4个单位，宽为3个单位，因此其面积为4 × 3 = 12 平方单位。

Perimeter = sum of all side lengths | 周长 = 所有边长之和

Area of rectangle = length × width | 长方形面积 = 长 × 宽

9. Drawing Polygons from Verbal Descriptions | 根据文字描述绘制多边形

Sometimes, we are asked to draw a polygon based on a description rather than explicit coordinates. For example: “Triangle ABC has a right angle at vertex B. Point A is at (1, 1), point B is at (1, 5), and AC is horizontal.”

有时，我们被要求根据文字描述而不是明确坐标来绘制多边形。例如：“三角形ABC的直角在顶点B处。点A在(1, 1)，点B在(1, 5)，且AC是水平的。”

Let us analyse this description. Since B is the right angle and AB is vertical (A and B share x = 1), the other side from B must be horizontal. Since AC is horizontal, both A and C must share the same y-coordinate (y = 1). Therefore, C must be at some point (c, 1) where c > 1 to make BC horizontal. If we choose C = (7, 1), then triangle ABC has a base AC of length 6 units and height AB of length 4 units.

让我们分析这个描述。由于B是直角且AB是垂直的（A和B都位于x = 1），从B出发的另一条边必须是水平的。由于AC是水平的，A和C必须有相同的y坐标（y = 1）。因此，C必须在某个点(c, 1)，其中c > 1使得BC为水平。如果我们选择C = (7, 1)，那么三角形ABC的底边AC长度为6个单位，高AB长度为4个单位。

Area of triangle ABC = ½ × 6 × 4 = 12 square units

三角形ABC的面积 = ½ × 6 × 4 = 12 平方单位

10. Common Mistakes and How to Avoid Them | 常见错误及避免方法

One common mistake is plotting (x, y) as (y, x), which reverses the point. To avoid this, remember the phrase “x is horizontal, y is vertical” and always plot the x-coordinate first. Another mistake is connecting points in the wrong order, which can create a self-intersecting shape instead of a simple polygon.

一个常见错误是将(x, y)错记为(y, x)，导致点被颠倒。为避免此问题，记住“x是水平方向，y是垂直方向”，且始终先描x坐标。另一个错误是连接点的顺序不对，这会产生自相交图形而不是简单多边形。

A third mistake is forgetting to close the polygon by connecting the last point back to the first. Always check your drawing: a polygon must be a closed figure. Finally, ensure all vertices are labelled clearly so that you can reference them in your calculations.

第三个错误是忘记将最后一个点连接回第一个点以封闭图形。始终检查你的图形：多边形必须是封闭图形。最后，确保所有顶点都清晰标注，以便后续计算时引用。

11. Practice Exercises | 练习巩固

Try drawing the following polygons on your own coordinate plane to reinforce your skills:

请在自己的坐标平面上尝试绘制以下多边形，以巩固所学技能：

Exercise 1: Rectangle with vertices at (−2, 1), (−2, 4), (5, 4), and (5, 1). Find its area.

练习1：顶点为(−2, 1)、(−2, 4)、(5, 4)和(5, 1)的长方形。求其面积。
Exercise 2: Triangle with vertices at (0, 0), (8, 0), and (4, 6). What type of triangle is it (based on side lengths)?

练习2：顶点为(0, 0)、(8, 0)和(4, 6)的三角形。按边长判断它是什么类型的三角形？
Exercise 3: Square with one vertex at (2, 3) and another at (2, 8). Find the coordinates of the other two vertices.

练习3：一个正方形，一个顶点为(2, 3)，另一个顶点为(2, 8)。求另外两个顶点的坐标。

For Exercise 1, the rectangle has a width of 7 units and a height of 3 units, so its area is 21 square units. For Exercise 2, since the base is 8 units and the apex is at x = 4 exactly above the midpoint, the triangle is isosceles. For Exercise 3, the side length is 5 units, so the other two vertices are (7, 3) and (7, 8), or (−3, 3) and (−3, 8), depending on which side of the segment you choose.

对于练习1，长方形宽为7个单位，高为3个单位，面积为21平方单位。对于练习2，底边为8个单位，顶点在x = 4即底边中点正上方，因此该三角形是等腰三角形。对于练习3，边长为5个单位，所以另外两个顶点为(7, 3)和(7, 8)，或(−3, 3)和(−3, 8)，取决于你选择线段的哪一侧。

12. Summary | 总结

Drawing polygons on the coordinate plane is a fundamental skill in geometry. We plot each vertex as an ordered pair (x, y), connect them in the correct order, and close the shape by connecting the last vertex back to the first. We can then analyse the shape by examining side lengths, parallel and perpendicular relationships, and calculate perimeter and area.

在坐标平面上绘制多边形是几何学中的一项基本技能。我们将每个顶点作为有序数对(x, y)描出，按正确顺序连接，并通过将最后一个顶点连接回第一个顶点来封闭图形。然后我们可以通过检查边长、平行和垂直关系来分析图形，并计算周长和面积。

Remember the key skills: plot points accurately with x first and y second; connect vertices in the given order; check that the shape is closed; and use coordinate differences to find horizontal and vertical side lengths. With practice, these techniques will become second nature.

记住关键技能：准确描点，x在前y在后；按给定顺序连接顶点；检查图形是否封闭；使用坐标差求水平和垂直边的长度。多加练习，这些技巧将变得得心应手。

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Drawing Polygons on the Coordinate Plane | 在坐标平面上绘制多边形