The main purpose of the distance formula is to measure the distance between two points using the Pythagoras Theorem. Without the distance formula, it is not always possible to measure the exact distance if a measuring tool like the ruler is not available. Instead, the distance formula makes use of relative coordinate points that are given to allow us to calculate the distance.

Let us consider an scenario where we are stranded in a desert and the only way out is traveling some distance along a certain direction.Now, the task is to find out the distance that we are required to travel given the coordinates of our starting and ending points.

We shall begin by stating out the distance formula :

where (x,y) are the respective points.

First we clearly indicate and label the points.

Taking the two points as diagonals,

we draw a right-angle triangle by connecting in the lines.

Next, we calculate the vertical (a) and horizontal (b) lengths of the right-angle triangle by simply subtracting the x and y values.

Finally, we make use of the Pythagoras theorem to determine the unknown hypotenuse (which is the distance we are looking for):

Scholar’s tips: Try not to mismatch the x and y values as this result in erroneous subtraction and hence a wrong answer. This is a common mistake that most students make and should be taken note of.

Always remember to place the square root sign throughout the application of the distance formula. This is because it is easy to accidentally omit the square root, causing the loss of easy marks.

Another points to remember is to work out everything within the brackets before proceeding with the squaring. Squaring is done on everything inside the brackets, including the negative sign, so the square of a negative will be positive. Basically, you are well advised to follow the order of operations and work it out step by step to improve your chances of obtaining the correct answer.

Most distance questions like their answers in exact form in surds. So instead of writing as 4.123 , it would be advisable to write unless otherwise stated in the question.

Example

Find the distance between (3,4) and (-2,3) in its exact form.

In this question, we simply have to apply the distance formula. Mark out the coordinates first and then draw out the right angle triangle. Then we applied the following distance formula :

Distance between the two points

=

=

=

Since the question asks for the exact form, we leave the answer in surds.

Questions:

1. How can the distance between two points be determined?

2. What must you know in order to use the Pythagorean Theorem to determine the distance between two points?

3. How do you find the length of the vertical leg?

4. How do you find the length of the horizontal leg?

5. Find the distance between (2,3) and ( -4, -5 ).

Midpoint Formula

Some coordinate geometry questions may require you to find the midpoint of line segments in the coordinate plane. To find a point that is halfway between two given points, get the average of the x-values and the average of the y-values.

The midpoint between the two points (x₁,y₁) and (x₂,y₂) is

For example:

The midpoint of the points A(1,4) and B(5,6) is
   

Questions:

1. Find the midpoint of the two points A(1, -3) and B(4, 5).

2. M(3, is the midpoint of the line AB. A has the coordinates (-2, 3), Find the coordinates of B.

Introduction:

So many things we use in maths don’t seem relevant in our “real” lives. Do you ever wonder why it will be faster to travel “as the crow flies” instead of walking the original paths?  About how tall a ladder should be so that the fireman can save the child from the burning building? All these questions can be solved by using the Pythagorean Theorem. It’s really as simple as a….  b….. c….

Questions

    Question 1…

    Who is Pythagoras?        

Pythagoras did lots of amazing things, but where did this man come from?
What was his big contribution to mathematics?
And what is this theorem used for?

 What is the Pythagorean Theorem?

Explain what the relationship is between the sides of a right triangle.

How can someone prove this theorem? 
            
( Research or look any methods or process that can show how to prove this theorem. If you able find so many ways,  choose one to explain).

How can this be used in “real life?”

See problems that use the Pythagorean Theorem, both from a text and using the samples
        in context.  When you are done studying their ideas, write one of your own.