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6th grade

Course: 6th grade > Unit 9

Lesson 3: Polygons on the coordinate plane
Math
6th grade
Coordinate plane
Polygons on the coordinate plane
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Coordinate plane FAQ

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Frequently asked questions about the coordinate plane

What is the coordinate plane?

The coordinate plane is a flat surface that has two number lines that cross at a point called the origin. The number line that goes left and right is called the x-axis, and the number line that goes up and down is called the y-axis. The x-axis and the y-axis divide the plane into four regions called quadrants. We label each quadrant with a Roman numeral in counterclockwise order: I, II, III, or IV.
A coordinate plane with an x-axis from negative 10 to positive 10 and a y-axis from negative 10 to positive 10.

How do we plot points on the coordinate plane?

Each point on the coordinate plane has a pair of numbers that tell us its position. These numbers are called coordinates, and we write them in parentheses, separated by a comma. The first number is the x-coordinate, which tells us how far left or right the point is from the origin. The second number is the y-coordinate, which tells us how far up or down the point is from the origin.
To plot a point on the coordinate plane, we need to know its coordinates. Then, we follow these steps:
  • Start at the origin, where the x-axis and the y-axis meet.
  • Move along the x-axis until you reach the x-coordinate of the point. If the x-coordinate is positive, move to the right. If the x-coordinate is negative, move to the left.
  • From there, move along a vertical line until you reach the y-coordinate of the point. If the y-coordinate is positive, move up. If the y-coordinate is negative, move down.
  • Mark the point with a dot and label it with its coordinates.
For example, to plot the point (3,−4), we start at the origin, move 3 units to the right along the x-axis. Then we move 4 units down along a vertical line. We mark the point with a dot and label it (3,−4).
A coordinate plane. An arrow starts at the origin and points right 3 units. A second arrow starts at the end of the first arrow and points down 4 units. There is a point at the end of the second arrow labeled (3, negative 4).

How do we measure horizontal and vertical distances on the coordinate plane?

Sometimes, we want to know how far apart two points are on the coordinate plane. This is called the distance between the points. One way to find the distance is to use a ruler or a string to measure the length of the straight line that connects the points. But what if we don't have a ruler or a string? Can we use math to find the distance?
Yes, we can! But first, we need to learn a special case of finding the distance: when the two points share an x- or y-coordinate. This means that they are either on the same horizontal line or on the same vertical line. For example, the points (3,7) and (3,−1) share an x-coordinate of 3, so they are on the same vertical line. The points (−2,4) and (−9,4) share a y-coordinate of 4, so they are on the same horizontal line.
Let's think about the distance between (3,7) and (3,−1). The point (3,7) is 7 units above the 3 on the x-axis. Is (3,−1) more or fewer than 7 units away from (3,7)? The point (3,−1) is 1 unit below the 3 on the x-axis. Since it is on the opposite side of the x-axis, it is 1 unit farther away.
7+1=8
So the distance between (3,7) and (3,−1) is 8 units.
Now let's consider the distance between the points (−2,4) and (−9,4). The point (−9,4) is 9 units to the left of the 4 on the y-axis. Is (−2,4) more or fewer than 9 units away from (−9,4)? The point (−2,4) is 2 units left of the 4 on the y-axis. Since it is on the same side of the y-axis, it is 2 units less distant from (−9,4).
9−2=7
So the distance between (−2,4) and (−9,4) is 7 units.

How do we find areas and perimeters of rectilinear figures on the coordinate plane?

First, we may need to draw the figures. If we know the coordinates of its vertices, which are the points where the sides meet, then, we follow these steps:
  • Plot the vertices on the coordinate plane using the coordinates.
  • Connect the vertices with straight lines in the order they are given. Make sure to close the shape by connecting the last vertex to the first vertex.
  • Label the polygon with its name and the coordinates of its vertices.
Sometimes we have to use what we know about the shapes to finish drawing them. For example, suppose we know the two bottom coordinates of a rectangle and its height. The opposite sides of a rectangle have equal lengths, and that the height is perpendicular to the base. We can use those facts to finish drawing the rectangle.
Then we find the lengths of the measurements.
For perimeter, we will need to find the length of every side, then add all of the lengths.
For the area of basic shapes, we often only need the length of the base and the height of the figure. We always measure the height at a perpendicular to the base. The area of a rectangle is the product of the length of the base times the height. For more complicated figures, we might need to split it into many rectangles. The total area will be the sum of the areas of all of the rectangles.

Why do we need to learn about the coordinate plane?

The coordinate plane is a useful tool for representing and analyzing many situations in the real world. For example, we can use the coordinate plane to:
  • Map locations and directions. The coordinates of a point can tell us its latitude and longitude, which are measures of its position on the globe. We can also use the coordinate plane to draw routes, calculate distances, and find bearings between places.
  • Graph equations and functions. The coordinates of a point can also tell us the values of a variable and its relationship to another variable. We can use the coordinate plane to plot points that satisfy an equation or a function, and see how the shape of the graph changes as we vary the variables.
  • Design and construct shapes and patterns. The coordinates of a point can also tell us its dimensions and orientation. We can use the coordinate plane to sketch and measure shapes and patterns, and apply transformations such as translations, rotations, and reflections.
  • Explore geometry and art. The coordinates of a point can also tell us its symmetry and beauty. We can use the coordinate plane to create and appreciate geometric and artistic figures, such as polygons, fractals, tessellations, and optical illusions.
As you can see, the coordinate plane is a very versatile and powerful tool that can help us understand and communicate many aspects of the world around us. We hope you enjoy learning about it and have fun with it!

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