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Distance Formula

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.

'Try it Yourself' Section

After going through this tutorial , you should be able to apply the distance formula.

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

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