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 , where is the , is the with its positive direction in the direction of , and line is the with its positive direction in the direction of . The length of line segment is the unit length of the , and that of is the unit length of the .
From this definition, we see that the orthonormal coordinate plane is a special coordinate plane since we have and .
Let us take three random noncollinear points , , and , which are positioned in space as shown.
If we want to use them to form the coordinate plane , we need to draw the lines and to form the - and and create a grid with lines parallel to both axes and spaced by the unit lengths defined by and .
We can observe that this coordinate plane is not orthonormal because and are not perpendicular. We call this type of coordinate plane an oblique coordinate plane.
An orthogonal coordinate plane is a coordinate plane in which and 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, and are not perpendicular.
In an orthogonal coordinate plane, .
In an orthonormal coordinate plane, and .
It is worth noting that, for the sake of convenience, we usually represent coordinate planes with a horizontal 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
is an isosceles triangle with a right angle at . The points ,
, and are the midpoints of the line segments ,
, and respectively.
- Which of the following planes is an orthonormal coordinate plane?
- Which of the following planes is an orthogonal but not an orthonormal coordinate plane?
- Which of the following planes is an oblique coordinate plane?
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 , and the line from the origin to the third point forms the .
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 . Triangle is right at , which means that is not a right angle. Lines and are therefore not perpendicular.
Option B is . As is a right angle, lines and are perpendicular.
Option C is . As is a right angle, lines and are perpendicular.
Option D is . Triangle is right at , which means that is not a right angle. Lines and are therefore not perpendicular.
Option E is . Triangle is right at , which means that is not a right angle. Lines and are therefore not perpendicular.
So, only in options B and C are the - and 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 , and for option C, it means whether .
As triangle is isosceles and has a right angle at , we have . Point is the midpoint of , and is that of , so . Hence, 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 . As is a right angle, lines and are perpendicular.
Option B is . As is a right angle, lines and are perpendicular.
Option C is . is the median of the isosceles triangle with ; it is therefore also the perpendicular bisector of . Hence, lines and are perpendicular.
Option D is . Triangle is right at , which means that is not a right angle. Lines and are therefore not perpendicular.
Option E is . As is a right angle, lines and 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 . As we have shown in part 1, we have .
Option B is . As we have shown in part 1, we have .
Option C is . Triangle is isosceles, so . Furthermore, is also the angle bisector of . Hence, . Thus, triangle is isosceles and .
Option E is . We have and , so .
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 is an oblique coordinate plane as lines and 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 .
Definition: Coordinates
Given a coordinate plane , the position of any point in the plane is described by its coordinates, noted .
is the real number on the of the point of intersection of the line parallel to the and going through .
is the real number on the of the point of intersection of the line parallel to the and going through .
Using this definition, we can determine the coordinates of a point in an oblique coordinate plane as shown in the following diagram.
It is worth noting that, by definition, the coordinates of in the coordinate plane are , and those of are .
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
and are two points in an orthonormal coordinate plane with a positive horizontal pointing to the right and a positive vertical pointing up. The unit lengths of the axes are given by the grid. If the coordinates of are , what are the coordinates of ?
Answer
We know that points and are in an orthonormal coordinate plane whose unit lengths are given by the grid. To find the coordinates of point , we will first determine the position of the origin of the coordinate plane, , using the coordinates of , . These coordinates mean that point 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 and 2 unit lengths down from .
We can now draw the axes of the coordinate plane and read the coordinates of point .
Point is one unit length left from the origin and is on the . Hence, its coordinates are .
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 , , , and in a coordinate plane.
- If the coordinate plane is an oblique coordinate plane, what is the shape of quadrilateral ?
- A trapezoid
- A kite
- A square
- A rectangle
- A parallelogram
- If the coordinate plane is an orthogonal coordinate plane, what is the shape of quadrilateral ?
- A trapezoid
- A kite
- A square
- A rectangle
- If the coordinate plane is an orthonormal coordinate plane, what is the shape of quadrilateral ?
- A trapezoid
- A kite
- A square
Answer
Part 1
Let us draw an oblique coordinate plane and place the points , , , and .
Since and on one hand and and on the other hand have the same -coordinate, we have . Similarly, and on one hand and and on the other hand have the same -coordinate, so . Hence, we can conclude that is a parallelogram (option E).
Part 2
Let us proceed like above and draw an orthogonal coordinate plane and place the points , , , and .
For the same reasons as in the oblique coordinate plane, is a parallelogram. However, it is a special parallelogram since the - and are perpendicular. A parallelogram with a right angle is a rectangle. Hence, is a rectangle (option D).
Part 3
Finally, for the orthonormal coordinate plane, we know now that . 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
is a parallelogram, and points , , , and are midpoints of the line segments , , , and respectively.
Find the coordinates of point in each of the following coordinate plane configurations.
- The coordinate plane
- The coordinate plane
- The coordinate plane
- The coordinate plane
Answer
Before starting, let us note that as is a parallelogram and the points , , , and are the midpoints of the line segments , , , and , we have and .
Part 1
In the coordinate plane , is the origin, is the with as its unit length and is the with as its unit length. Hence, the coordinates of are and those of are .
To find the -coordinate of point , we look for the line parallel to the (line ) that goes through ; it is . It intersects the (line ) at , corresponding to an -coordinate of 1.
To find the -coordinate of point , we look for the line parallel to the (line ) that goes through ; it is . It intersects the (line ) at , corresponding to a -coordinate of 1. The coordinates of in are .
Part 2
In the coordinate plane , is the origin. Its coordinates are therefore .
Part 3
In the coordinate plane , is the origin, is the with as its unit length, and is the with as its unit length. Hence, the coordinates of are and those of are .
The line parallel to the going through is . It intersects the at , giving an -coordinate of 1.
The line parallel to the going through is . It intersects the at , giving a -coordinate of 1.
Hence, the coordinates of in are .
Part 4
In the coordinate plane , is the origin, is the with as its unit length, and is the with as its unit length. Hence, the coordinates of are and those of are .
The line parallel to the going through is . It intersects the at . being the midpoint of , we have , which corresponds to an -coordinate of 2.
The line parallel to the going through is . It intersects the at . Since is the midpoint of , we have , which corresponds to a -coordinate of 2.
Hence, the coordinates of in are .
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 , , and
in the orthonormal coordinate plane .
- What type of coordinate plane is ?
- What are the coordinates of point in the coordinate plane ?
Answer
Part 1
In the coordinate plane , is the origin, is the with as its unit length, and is the with as its unit length. To identify the type of coordinate plane, we need to determine
- whether and are perpendicular,
- whether .
We note that both and 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 and . Hence, as , we find that .
In addition, as the diagonals of a square are of equal lengths and and are diagonals of two congruent squares, we have . Thus, we can conclude that is an orthonormal coordinate plane since its axes are perpendicular and they have the same unit length.
Part 2
To find the coordinates of in , we need to draw the two lines parallel to the - and going through .
The line parallel to the going through intersects the at , giving an -coordinate of 1.
The line parallel to the going through intersects the at a point that is a distance from the origin twice the length (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 (on the same side as ). Hence, it corresponds to a -coordinate of 2.
Hence, the coordinates of in are .
Let us finish by recapping some key points from the explainer.
Key Points
- A coordinate plane is formed by any three noncollinear points , where is the , is the with its positive direction in the direction of , and line is the with its positive direction in the direction of . The length of line segment is the unit length of the , and that of is the unit length of the .
- 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 and are not perpendicular, orthogonal, where and are perpendicular, and orthonormal, which is an orthogonal plane with the added condition that .
- Given a coordinate plane , the position of any point in the plane is described by its coordinates, noted . is the real number on the of the point of intersection of the line parallel to the and going through . is the real number on the of the point of intersection of the line parallel to the and going through .
