What is a Plane?

Planes in maths are quite common. In fact, they form the basis of most upper-level mathematics courses. In order to find the distance between two points, let’s make sure to first cover the basics.

 

graph_quadrants
A plane is divided into four quadrants. It is divided by two axes: the x-axis and the y-axis. Depending on which quadrant your point is in, your point will exhibit different behaviour. 

Quadrant X-axis Y-axis Coordinates
A I Positive Positive (+,+)
B II Negative Positive (-,+)
C III Negative Negative (-,-)
D IV Positive Negative (+,+)

 

The best Maths tutors available
Intasar
4.9
4.9 (36 reviews)
Intasar
£48
/h
Gift icon
1st lesson free!
Paolo
4.9
4.9 (29 reviews)
Paolo
£30
/h
Gift icon
1st lesson free!
Jamie
5
5 (16 reviews)
Jamie
£25
/h
Gift icon
1st lesson free!
Harinder
5
5 (16 reviews)
Harinder
£15
/h
Gift icon
1st lesson free!
Sehaj
5
5 (32 reviews)
Sehaj
£25
/h
Gift icon
1st lesson free!
Matthew
5
5 (17 reviews)
Matthew
£30
/h
Gift icon
1st lesson free!
Gian lorenzo
4.9
4.9 (8 reviews)
Gian lorenzo
£65
/h
Gift icon
1st lesson free!
Petar
4.9
4.9 (12 reviews)
Petar
£40
/h
Gift icon
1st lesson free!
Intasar
4.9
4.9 (36 reviews)
Intasar
£48
/h
Gift icon
1st lesson free!
Paolo
4.9
4.9 (29 reviews)
Paolo
£30
/h
Gift icon
1st lesson free!
Jamie
5
5 (16 reviews)
Jamie
£25
/h
Gift icon
1st lesson free!
Harinder
5
5 (16 reviews)
Harinder
£15
/h
Gift icon
1st lesson free!
Sehaj
5
5 (32 reviews)
Sehaj
£25
/h
Gift icon
1st lesson free!
Matthew
5
5 (17 reviews)
Matthew
£30
/h
Gift icon
1st lesson free!
Gian lorenzo
4.9
4.9 (8 reviews)
Gian lorenzo
£65
/h
Gift icon
1st lesson free!
Petar
4.9
4.9 (12 reviews)
Petar
£40
/h
Gift icon
1st lesson free!
Let's go

Point Definition

A point on a plane is defined by coordinates. You can think of coordinates as a set of directions to arrive to a point on a plane.

 

Definition Standard Form Example
Coordinates A set of numbers that help you place a point on a plane (x coordinate, y coordinate) (2,-5)

 

Let’s take a look at an example.

direction_on_x_axis

If we didn’t have a standard form for coordinates, we could say the following:

 

Directions
A Move 5 places to the right, go up 2 spaces
B Move 3 spaces to the left, go 1 space down

 

Instead of having to give these directions all the time, we can just give the coordinates of a point. Keep in mind that both the x and y coordinates only move in two directions:

 

Coordinate Possible Directions Positive Direction Negative Direction
x Left or Right Right Left
y Up or Down Up Down

 

Formula Distance Between Two Points

When you want to find the distance between two points, you simply follow the formula below.

distance_two_points

Here, you can see what the formula means.

 

Coordinates X Y
Point 1 (x_{1},y_{1}) x_{1} y_{1}
Point 2 (x_{2},y_{2}) x_{2} y_{2}

 

Example 1

Let’s go through an example together. Take the following two points.

two_points_graph

 

Standard Forms Coordinates
Point A (x_{1},y_{1}) (2,2)
Point B (x_{2},y_{2}) (5,8)

 

To make it easier, let’s split up the coordinates like we did in the previous section. This way, we can simply plug it into our formula.

 

x_{1} = 2
x_{2} = 5
y_{1} = 2
y_{2} = 8

 

In order to find the distance, we simply need to plug it into the distance formula.

distance_formula

 

Step 1 Plug in the coordinates sqrt{5 - 2)^{2} + (8 - 2)^{2}}
Step 2 Simplify the equation sqrt{9 + 36}
Step 3 Take the square root sqrt{45} = 6.7

 

Pythagorean Theorem

You may be wondering if there’s an easier way to find the distance between two points. Another way to find the distance between two points is if you have a right triangle on the graph.

right_triangle_graph

In this case, you can use the Pythagorean theorem. Take a look at the information below.

 

Right triangle To use the Pythagorean theorem, you must have a right triangle, which is a triangle whose largest angle is 90 degrees.
Two sides You must know the length of at least two sides in order to find the length of the last side.
Hypotenuse The hypotenuse (c) is the largest side in a right triangle. The hypotenuse is always the side that is opposite the right angle.

 

In this case, you can use the Pythagorean theorem as long as you have the length of two sides.

 

right_triangle_pythagorean
Pythagorean Theorem a^{2} + b^{2} = c^{2}

 

Problem 1

Find the distance between the origin of the graph and the point given below.

 

Coordinates
Point A (4,5)

 

Problem 2

On a graph, you have a square made up of four points. You know two of those points - find the length of the diagonal. Next, find area of the triangle formed by the square’s diagonal.

 

right_triangle_points
Point 1 (8,5)
Point 2 (8,1)

 

Solution Problem 1

In order to find the distance from the origin to the point, you should know what the origin is. Recall that the origin is located in the middle of the graph.

 

Coordinates
Origin (0,0)
Point A (4,5)

 

So, to find the distance from the point given to the graph, we simply need to plug the two points into the distance formula.

 

Step 1 Get the distance formula sqrt{x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}
Step 2 Plug in the points sqrt{0 - 4)^{2} + (0 - 5)^{2}}
Step 3 Solve the equation sqrt{41} = 6.4

 

The length between the origin and the point is 6.4.

 

Solution Problem 2

In this problem, we need to find the area of the square. However, we only have the coordinates of one side. To solve the problem, you should recall a couple of facts.

 

Fact 1 All the sides of a square have the same length
Fact 2 A square is made up of four right angles
Fact 3 Because of these right angles, the diagonal of a square makes a hypotenuse

 

So, the first step is to find the distance between the two points we’re given: (8,1) and (8,5).

 

Step 1 Get the distance formula sqrt{x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}
Step 2 Plug in the points sqrt{8 - 8)^{2} + (5 - 1)^{2}}
Step 3 Solve the equation sqrt{16} = 4

 

Because all sides of a square are the same length, we know that a = b.

square_points_right_triangle

Recall the formula for the area of a triangle.

 

Step 1 Formula for the area of a triangle frac{1}{2}*b*h
Step 2 Plug in the base (b) and the height (h) of the triangle frac{1}{2}*4*4
Step 3 Solve the equation 8

 

The area of the triangle is 8.

 

Finally, because we’re dealing with a right triangle, we can simply use the Pythagorean to discover the length of the diagonal.

 

Step 1 Get the Pythagorean formula a^{2} + b^{2} = c^{2}
Step 2 Plug in the points 4^{2} + 4^{2} = c^{2}
Step 3 Solve the equation 32 = c^{2}
Step 4 Take the square root sqrt{32} = c = 5.7

 

>

The platform that connects tutors and students

Did you like this article? Rate it!

1 Star2 Stars3 Stars4 Stars5 Stars 4.00 (3 rating(s))
Loading...

Danica

Located in Prague and studying to become a Statistician, I enjoy reading, writing, and exploring new places.