Lesson Explainer: Coordinate Planes Mathematics

In this explainer, we will learn how to define the different types of coordinate planes and the coordinates of a point and place points on the plane.

Coordinate planes are particularly useful to locate objects using their coordinates. On Earth, we commonly use the geographic coordinate system (GCS) based on latitude and longitude. In geometry, we generally use a coordinate plane where the axes are perpendicular and the spacings are equal. In this explainer, we will look more at this familiar coordinate plane, along with some alternative coordinate planes.

Let us first define a coordinate plane in general terms.

Definition: Coordinate Plane

A coordinate plane is formed by any three noncollinear points (𝑂;𝐼,𝐽), where 𝑂 is the origin, line ⃖⃗𝑂𝐼 is the π‘₯-axis with its positive direction in the direction of οƒͺ𝑂𝐼, and line ⃖⃗𝑂𝐽 is the 𝑦-axis with its positive direction in the direction of οƒͺ𝑂𝐽. The length of line segment 𝑂𝐼 is the unit length of the π‘₯-axis, and that of 𝑂𝐽 is the unit length of the 𝑦-axis.

From this definition, we see that the orthonormal coordinate plane is a special coordinate plane since we have βƒ–οƒ©οƒ©οƒ©βƒ—π‘‚πΌβŸ‚βƒ–οƒ©οƒ©οƒ©βƒ—π‘‚π½ and 𝑂𝐼=𝑂𝐽.

Let us take three random noncollinear points 𝑂, 𝐼, and 𝐽, which are positioned in space as shown.

If we want to use them to form the coordinate plane (𝑂;𝐼,𝐽), we need to draw the lines ⃖⃗𝑂𝐼 and ⃖⃗𝑂𝐽 to form the π‘₯- and 𝑦-axes and create a grid with lines parallel to both axes and spaced by the unit lengths defined by 𝑂𝐼 and 𝑂𝐽.

We can observe that this coordinate plane is not orthonormal because ⃖⃗𝑂𝐼 and ⃖⃗𝑂𝐽 are not perpendicular. We call this type of coordinate plane an oblique coordinate plane.

An orthogonal coordinate plane is a coordinate plane in which ⃖⃗𝑂𝐼 and ⃖⃗𝑂𝐽 are perpendicular.

Therefore, we have three different types of coordinate planes that we will outline now.

Definition: Types of Coordinate Planes (𝑂; 𝐼, 𝐽)

In an oblique coordinate plane, ⃖⃗𝑂𝐼 and ⃖⃗𝑂𝐽 are not perpendicular.

In an orthogonal coordinate plane, βƒ–οƒ©οƒ©οƒ©βƒ—π‘‚πΌβŸ‚βƒ–οƒ©οƒ©οƒ©βƒ—π‘‚π½.

In an orthonormal coordinate plane, βƒ–οƒ©οƒ©οƒ©βƒ—π‘‚πΌβŸ‚βƒ–οƒ©οƒ©οƒ©βƒ—π‘‚π½ and 𝑂𝐼=𝑂𝐽.

It is worth noting that, for the sake of convenience, we usually represent coordinate planes with a horizontal π‘₯-axis whenever possible as this makes visual interpretation easier.

In our first example, we are going to use these definitions to identify different types of coordinate planes.

Example 1: Identifying Orthonormal, Orthogonal, and Oblique Coordinate Planes

𝐴𝐡𝐢 is an isosceles triangle with a right angle at 𝐡. The points 𝐷, 𝐸, and 𝐹 are the midpoints of the line segments 𝐴𝐢, 𝐴𝐡, and 𝐡𝐢 respectively.

  1. Which of the following planes is an orthonormal coordinate plane?
    1. (𝐴;𝐸,𝐷)
    2. (𝐡;𝐢,𝐸)
    3. (𝐡;𝐹,𝐸)
    4. (𝐴;𝐡,𝐢)
    5. (𝐢;𝐴,𝐡)
  2. Which of the following planes is an orthogonal but not an orthonormal coordinate plane?
    1. (𝐡;𝐹,𝐸)
    2. (𝐡;𝐢,𝐴)
    3. (𝐷;𝐡,𝐢)
    4. (𝐴;𝐡,𝐢)
    5. (𝐡;𝐹,𝐴)
  3. Which of the following planes is an oblique coordinate plane?
    1. (𝐷;𝐡,𝐢)
    2. (𝐡;𝐢,𝐷)
    3. (𝐡;𝐢,𝐴)
    4. (𝐷;𝐡,𝐴)
    5. (𝐸;𝐡,𝐷)

Answer

Part 1

Recall that the first point given for the coordinate plane is its origin. The line from the origin to the second point forms the π‘₯-axis, and the line from the origin to the third point forms the 𝑦-axis.

In an orthonormal coordinate plane, the two axes are perpendicular and the length units, defined as the distances between the origin and the second and third points respectively, are equal.

Let us go through all the options and evaluate if the first criterion (the axes are perpendicular) is met.

Option A is (𝐴;𝐸,𝐷). Triangle 𝐴𝐡𝐢 is right at 𝐡, which means that ∠𝐡𝐴𝐢 is not a right angle. Lines ⃖⃗𝐴𝐸 and ⃖⃗𝐴𝐷 are therefore not perpendicular.

Option B is (𝐡;𝐢,𝐸). As ∠𝐴𝐡𝐢 is a right angle, lines ⃖⃗𝐡𝐢 and ⃖⃗𝐡𝐸 are perpendicular.

Option C is (𝐡;𝐹,𝐸). As ∠𝐴𝐡𝐢 is a right angle, lines ⃖⃗𝐡𝐹 and ⃖⃗𝐡𝐸 are perpendicular.

Option D is (𝐴;𝐡,𝐢). Triangle 𝐴𝐡𝐢 is right at 𝐡, which means that ∠𝐡𝐴𝐢 is not a right angle. Lines ⃖⃗𝐴𝐡 and ⃖⃗𝐴𝐢 are therefore not perpendicular.

Option E is (𝐢;𝐴,𝐡). Triangle 𝐴𝐡𝐢 is right at 𝐡, which means that ∠𝐡𝐢𝐴 is not a right angle. Lines ⃖⃗𝐢𝐡 and ⃖⃗𝐢𝐴 are therefore not perpendicular.

So, only in options B and C are the π‘₯- and 𝑦-axes perpendicular. We need now to assess for each of them whether the second criterion is met (the unit lengths of both axes are equal). For option B, it means whether 𝐡𝐢=𝐡𝐸, and for option C, it means whether 𝐡𝐹=𝐡𝐸.

As triangle 𝐴𝐡𝐢 is isosceles and has a right angle at 𝐡, we have 𝐡𝐢=𝐡𝐴. Point 𝐸 is the midpoint of 𝐴𝐡, and 𝐹 is that of 𝐡𝐢, so 𝐡𝐹=𝐡𝐸. Hence, (𝐡;𝐹,𝐸) is the only orthonormal coordinate plane. This is option C.

Part 2

Recall that, in an orthogonal coordinate plane, the two axes are perpendicular. We have also been told that the coordinate planes are not orthonormal, meaning that the lengths between the origin and the second and third points, respectively, are not equal.

As in part 1, let us go through the options and eliminate first the coordinate planes whose axes are not perpendicular.

Option A is (𝐡;𝐹,𝐸). As ∠𝐴𝐡𝐢 is a right angle, lines ⃖⃗𝐡𝐹 and ⃖⃗𝐡𝐸 are perpendicular.

Option B is (𝐡;𝐢,𝐴). As ∠𝐴𝐡𝐢 is a right angle, lines ⃖⃗𝐡𝐢 and ⃖⃗𝐡𝐴 are perpendicular.

Option C is (𝐷;𝐡,𝐢). 𝐡𝐷 is the median of the isosceles triangle 𝐴𝐡𝐢 with 𝐡𝐢=𝐡𝐴; it is therefore also the perpendicular bisector of 𝐴𝐢. Hence, lines ⃖⃗𝐷𝐡 and ⃖⃗𝐷𝐢 are perpendicular.

Option D is (𝐴;𝐡,𝐢). Triangle 𝐴𝐡𝐢 is right at 𝐡, which means that ∠𝐡𝐴𝐢 is not a right angle. Lines ⃖⃗𝐴𝐡 and ⃖⃗𝐴𝐢 are therefore not perpendicular.

Option E is (𝐡;𝐹,𝐴). As ∠𝐴𝐡𝐢 is a right angle, lines ⃖⃗𝐡𝐢 and ⃖⃗𝐡𝐴 are perpendicular.

Only option D is eliminated since its axes are not perpendicular, so it cannot be an orthogonal coordinate system.

Let us now compare the unit lengths of both axes.

Option A is (𝐡;𝐹,𝐸). As we have shown in part 1, we have 𝐡𝐹=𝐡𝐸.

Option B is (𝐡;𝐢,𝐴). As we have shown in part 1, we have 𝐡𝐢=𝐡𝐴.

Option C is (𝐷;𝐡,𝐢). Triangle 𝐴𝐡𝐢 is isosceles, so ∠𝐡𝐴𝐢=∠𝐡𝐢𝐴=45∘. Furthermore, 𝐡𝐷 is also the angle bisector of ∠𝐴𝐡𝐢. Hence, ∠𝐢𝐡𝐷=902=45∘. Thus, triangle 𝐡𝐢𝐷 is isosceles and 𝐷𝐡=𝐷𝐢.

Option E is (𝐡;𝐹,𝐴). We have 𝐡𝐹=12𝐡𝐴 and 𝐡𝐢=𝐡𝐴, so 𝐡𝐹≠𝐡𝐴.

(𝐡;𝐹,𝐴) is therefore an orthogonal coordinate plane. This is option E.

Part 3

We need to identify which coordinate plane is oblique, which means the one that has nonperpendicular axes.

Examining all options, we see that only (𝐡;𝐢,𝐷) is an oblique coordinate plane as lines ⃖⃗𝐡𝐢 and ⃖⃗𝐡𝐷 are not perpendicular. This is option B.

Now that we have defined these three different types of coordinate planes, let us define coordinates in a coordinate plane (𝑂;𝐼,𝐽).

Definition: Coordinates

Given a coordinate plane (𝑂;𝐼,𝐽), the position of any point 𝑀 in the plane is described by its coordinates, noted (π‘₯,𝑦).

π‘₯ is the real number on the π‘₯-axis of the point of intersection of the line parallel to the 𝑦-axis and going through 𝑀.

π‘¦οŒ¬ is the real number on the 𝑦-axis of the point of intersection of the line parallel to the π‘₯-axis and going through 𝑀.

Using this definition, we can determine the coordinates (π‘₯,𝑦) of a point 𝑀 in an oblique coordinate plane as shown in the following diagram.

It is worth noting that, by definition, the coordinates of 𝐼 in the coordinate plane (𝑂;𝐼,𝐽) are (1,0), and those of 𝐽 are (0,1).

In our next example, we are going to use our understanding of coordinates in an orthonormal coordinate plane.

Example 2: Determining the Coordinates of a Point given the Coordinates of Another Point in a Coordinate Plane

𝐴 and 𝐡 are two points in an orthonormal coordinate plane with a positive horizontal π‘₯-axis pointing to the right and a positive vertical 𝑦-axis pointing up. The unit lengths of the axes are given by the grid. If the coordinates of 𝐴 are (1,2), what are the coordinates of 𝐡?

Answer

We know that points 𝐴 and 𝐡 are in an orthonormal coordinate plane whose unit lengths are given by the grid. To find the coordinates of point 𝐡, we will first determine the position of the origin of the coordinate plane, 𝑂, using the coordinates of 𝐴, (1,2). These coordinates mean that point 𝐴 is located one unit length to the right of the origin and 2 unit lengths up from the origin. In other words, the origin of the coordinate plane is one unit length left from 𝐴 and 2 unit lengths down from 𝐴.

We can now draw the axes of the coordinate plane and read the coordinates of point 𝐡.

Point 𝐡 is one unit length left from the origin and is on the π‘₯-axis. Hence, its coordinates are (βˆ’1,0).

In the previous example, we were considering points in an orthonormal coordinate system. With the next example, we will reflect on the differences between the three types of coordinate planes.

Example 3: Determining the Shape of a Quadrilateral in Different Coordinate Plane Types

Consider the points 𝐴(0,0), 𝐡(1,0), 𝐢(1,1), and 𝐷(0,1) in a coordinate plane.

  1. If the coordinate plane is an oblique coordinate plane, what is the shape of quadrilateral 𝐴𝐡𝐢𝐷?
    1. A trapezoid
    2. A kite
    3. A square
    4. A rectangle
    5. A parallelogram
  2. If the coordinate plane is an orthogonal coordinate plane, what is the shape of quadrilateral 𝐴𝐡𝐢𝐷?
    1. A trapezoid
    2. A kite
    3. A square
    4. A rectangle
  3. If the coordinate plane is an orthonormal coordinate plane, what is the shape of quadrilateral 𝐴𝐡𝐢𝐷?
    1. A trapezoid
    2. A kite
    3. A square

Answer

Part 1

Let us draw an oblique coordinate plane and place the points 𝐴, 𝐡, 𝐢, and 𝐷.

Since 𝐴 and 𝐷 on one hand and 𝐡 and 𝐢 on the other hand have the same π‘₯-coordinate, we have 𝐴𝐷⫽𝐡𝐢⫽𝑦-axis. Similarly, 𝐴 and 𝐡 on one hand and 𝐷 and 𝐢 on the other hand have the same 𝑦-coordinate, so 𝐴𝐡⫽𝐷𝐢⫽π‘₯-axis. Hence, we can conclude that 𝐴𝐡𝐢𝐷 is a parallelogram (option E).

Part 2

Let us proceed like above and draw an orthogonal coordinate plane and place the points 𝐴, 𝐡, 𝐢, and 𝐷.

For the same reasons as in the oblique coordinate plane, 𝐴𝐡𝐢𝐷 is a parallelogram. However, it is a special parallelogram since the π‘₯- and 𝑦-axes are perpendicular. A parallelogram with a right angle is a rectangle. Hence, 𝐴𝐡𝐢𝐷 is a rectangle (option D).

Part 3

Finally, for the orthonormal coordinate plane, we know now that 𝐴𝐡=𝐴𝐷. 𝐴𝐡𝐢𝐷 is therefore a special rectangle where all sides are equal; it is a square (option C).

Let us practice with our next example reading coordinates in an oblique coordinate plane, keeping in mind that the grid in an oblique coordinate plane forms parallelograms.

Example 4: Finding the Coordinates of a Point in Multiple Configurations for the Coordinate Plane

𝐴𝐡𝐢𝐷 is a parallelogram, and points 𝐼, 𝐽, 𝐾, and 𝐿 are midpoints of the line segments 𝐴𝐡, 𝐡𝐢, 𝐢𝐷, and 𝐷𝐴 respectively.

Find the coordinates of point 𝐢 in each of the following coordinate plane configurations.

  1. The coordinate plane (𝐴;𝐡,𝐷)
  2. The coordinate plane (𝐢;𝐷,𝐡)
  3. The coordinate plane (𝐿;𝐽,𝐷)
  4. The coordinate plane (𝐴;𝐼,𝐿)

Answer

Before starting, let us note that as 𝐴𝐡𝐢𝐷 is a parallelogram and the points 𝐼, 𝐽, 𝐾, and 𝐿 are the midpoints of the line segments 𝐴𝐡, 𝐡𝐢, 𝐢𝐷, and 𝐷𝐴, we have ⃖⃗𝐴𝐡⫽⃖⃗𝐷𝐢⫽⃖⃗𝐿𝐽 and ⃖⃗𝐴𝐷⫽⃖⃗𝐡𝐢⫽⃖⃗𝐼𝐾.

Part 1

In the coordinate plane (𝐴;𝐡,𝐷), 𝐴 is the origin, ⃖⃗𝐴𝐡 is the π‘₯-axis with 𝐴𝐡 as its unit length and ⃖⃗𝐴𝐷 is the 𝑦-axis with 𝐴𝐷 as its unit length. Hence, the coordinates of 𝐡 are (1,0) and those of 𝐷 are (0,1).

To find the π‘₯-coordinate of point 𝐢, we look for the line parallel to the 𝑦-axis (line ⃖⃗𝐴𝐷) that goes through 𝐢; it is ⃖⃗𝐡𝐢. It intersects the π‘₯-axis (line ⃖⃗𝐴𝐡) at 𝐡, corresponding to an π‘₯-coordinate of 1.

To find the 𝑦-coordinate of point 𝐢, we look for the line parallel to the π‘₯-axis (line ⃖⃗𝐴𝐡) that goes through 𝐢; it is ⃖⃗𝐷𝐢. It intersects the 𝑦-axis (line ⃖⃗𝐴𝐷) at 𝐷, corresponding to a 𝑦-coordinate of 1. The coordinates of 𝐢 in (𝐴;𝐡,𝐷) are (1,1).

Part 2

In the coordinate plane (𝐢;𝐷,𝐡), 𝐢 is the origin. Its coordinates are therefore (0,0).

Part 3

In the coordinate plane (𝐿;𝐽,𝐷), 𝐿 is the origin, ⃖⃗𝐿𝐽 is the π‘₯-axis with 𝐿𝐽 as its unit length, and ⃖⃗𝐿𝐷 is the 𝑦-axis with 𝐿𝐷 as its unit length. Hence, the coordinates of 𝐽 are (1,0) and those of 𝐷 are (0,1).

The line parallel to the 𝑦-axis ⃖⃗𝐿𝐷 going through 𝐢 is ⃖⃗𝐽𝐢. It intersects the π‘₯-axis ⃖⃗𝐿𝐽 at 𝐽, giving an π‘₯-coordinate of 1.

The line parallel to the π‘₯-axis ⃖⃗𝐿𝐽 going through 𝐢 is ⃖⃗𝐷𝐢. It intersects the 𝑦-axis ⃖⃗𝐿𝐷 at 𝐷, giving a 𝑦-coordinate of 1.

Hence, the coordinates of 𝐢 in (𝐿;𝐽,𝐷) are (1,1).

Part 4

In the coordinate plane (𝐴;𝐼,𝐿), 𝐴 is the origin, ⃖⃗𝐴𝐼 is the π‘₯-axis with 𝐴𝐼 as its unit length, and ⃖⃗𝐴𝐿 is the 𝑦-axis with 𝐴𝐿 as its unit length. Hence, the coordinates of 𝐼 are (1,0) and those of 𝐿 are (0,1).

The line parallel to the 𝑦-axis ⃖⃗𝐴𝐿 going through 𝐢 is ⃖⃗𝐡𝐢. It intersects the π‘₯-axis ⃖⃗𝐴𝐼 at 𝐡. 𝐼 being the midpoint of 𝐴𝐡, we have 𝐴𝐡=2𝐴𝐼, which corresponds to an π‘₯-coordinate of 2.

The line parallel to the π‘₯-axis ⃖⃗𝐴𝐼 going through 𝐢 is ⃖⃗𝐷𝐢. It intersects the 𝑦-axis ⃖⃗𝐴𝐿 at 𝐷. Since 𝐿 is the midpoint of 𝐴𝐷, we have 𝐴𝐷=2𝐴𝐿, which corresponds to a 𝑦-coordinate of 2.

Hence, the coordinates of 𝐢 in (𝐴;𝐼,𝐿) are (2,2).

In our last example, we will have to determine the coordinates of a given point in a newly defined coordinate plane.

Example 5: Determining the Type of a Given Coordinate Plane and the Coordinates of a Point in a Different Coordinate Plane

Consider points 𝐴(1,1), 𝐡(βˆ’1,1), and 𝐢(βˆ’1,3) in the orthonormal coordinate plane (𝑂;𝐼,𝐽).

  1. What type of coordinate plane is (𝑂;𝐴,𝐡)?
  2. What are the coordinates of point 𝐢 in the coordinate plane (𝑂;𝐴,𝐡)?

Answer

Part 1

In the coordinate plane (𝑂;𝐴,𝐡), 𝑂 is the origin, ⃖⃗𝑂𝐴 is the π‘₯-axis with 𝑂𝐴 as its unit length, and ⃖⃗𝑂𝐡 is the 𝑦-axis with 𝑂𝐡 as its unit length. To identify the type of coordinate plane, we need to determine

  1. whether ⃖⃗𝑂𝐴 and ⃖⃗𝑂𝐡 are perpendicular,
  2. whether 𝑂𝐴=𝑂𝐡.

We note that both 𝑂𝐴 and 𝑂𝐡 are diagonals of a square of the grid. We know that the diagonal of a square is an axis of symmetry of the square, which means that π‘šβˆ π΄π‘‚πΌ=π‘šβˆ π΄π‘‚π½=45∘ and π‘šβˆ π΅π‘‚π½=45∘. Hence, as π‘šβˆ π΄π‘‚π΅=π‘šβˆ π΄π‘‚π½+π‘šβˆ π½π‘‚π΅, we find that π‘šβˆ π΄π‘‚π΅=45+45=90∘.

In addition, as the diagonals of a square are of equal lengths and 𝑂𝐴 and 𝑂𝐡 are diagonals of two congruent squares, we have 𝑂𝐴=𝑂𝐡. Thus, we can conclude that (𝑂;𝐴,𝐡) is an orthonormal coordinate plane since its axes are perpendicular and they have the same unit length.

Part 2

To find the coordinates of 𝐢 in (𝑂;𝐴,𝐡), we need to draw the two lines parallel to the π‘₯- and 𝑦-axes going through 𝐢.

The line parallel to the 𝑦-axis going through 𝐢 intersects the π‘₯-axis at 𝐴, giving an π‘₯-coordinate of 1.

The line parallel to the π‘₯-axis going through 𝐢 intersects the 𝑦-axis at a point that is a distance from the origin twice the length 𝑂𝐡 (the line segment from the origin to this intersection point is twice the diagonal of a grid square) and on the positive side of the 𝑦-axis (on the same side as 𝐡). Hence, it corresponds to a 𝑦-coordinate of 2.

Hence, the coordinates of 𝐢 in (𝑂;𝐴,𝐡) are (1,2).

Let us finish by recapping some key points from the explainer.

Key Points

  • A coordinate plane is formed by any three noncollinear points (𝑂;𝐼,𝐽), where 𝑂 is the origin, line ⃖⃗𝑂𝐼 is the π‘₯-axis with its positive direction in the direction of οƒͺ𝑂𝐼, and line ⃖⃗𝑂𝐽 is the 𝑦-axis with its positive direction in the direction of οƒͺ𝑂𝐽. The length of line segment 𝑂𝐼 is the unit length of the π‘₯-axis, and that of 𝑂𝐽 is the unit length of the 𝑦-axis.
  • The standard coordinate system that we use in mathematics is called an orthonormal coordinate system, but there are three main types of coordinate planes: oblique, where 𝑂𝐼 and 𝑂𝐽 are not perpendicular, orthogonal, where 𝑂𝐼 and 𝑂𝐽 are perpendicular, and orthonormal, which is an orthogonal plane with the added condition that 𝑂𝐼=𝑂𝐽.
  • Given a coordinate plane (𝑂;𝐼,𝐽), the position of any point 𝑀 in the plane is described by its coordinates, noted (π‘₯,𝑦). π‘₯ is the real number on the π‘₯-axis of the point of intersection of the line parallel to the 𝑦-axis and going through 𝑀. π‘¦οŒ¬ is the real number on the 𝑦-axis of the point of intersection of the line parallel to the π‘₯-axis and going through 𝑀.

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