Distance Between Two Points Revision Notes for Grade 10 NSC Matric Mathematics

Distance Between Two Points

What is a point and distance?

Understanding the basics is essential before we can calculate distances between points.

Point: A point represents a specific location on a coordinate plane. It is written as an ordered pair of numbers in the form $(x, y)$ , where $x$ is the horizontal position and $y$ is the vertical position.

Distance: Distance measures how far apart two points are from each other. It is always a positive value representing the length of the straight line connecting the two points.

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When working with coordinate geometry, always remember that points are precise locations defined by their x and y coordinates, while distance represents the actual measurement between any two such locations.

Deriving the distance formula

To find a general formula for calculating distance between any two points, we use the Pythagoras theorem. This involves creating a right triangle between the two points.

Consider two points $A(x_1; y_1)$ and $B(x_2; y_2)$ . We can form a right triangle by adding a third point $C(x_2; y_1)$ .

Using the Pythagoras theorem: $AB^2 = AC^2 + BC^2$

Therefore: $AB = \sqrt{AC^2 + BC^2}$

The horizontal distance is: $AC = x_2 - x_1$

The vertical distance is: $BC = y_2 - y_1$

Substituting these values: $AB = \sqrt{AC^2 + BC^2} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

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Key Insight: By forming a right triangle between any two points, we can always apply Pythagoras' theorem to find the direct distance. This works because the horizontal and vertical distances form the two shorter sides of the triangle, while the distance between the points is the hypotenuse.

The distance formula

For any two points $(x_1; y_1)$ and $(x_2; y_2)$ , the distance formula is:

$d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$

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Essential Property: $(x_1 - x_2)^2 = (x_2 - x_1)^2$ , so the order of subtraction doesn't matter since we square the difference. This means you can subtract the coordinates in either direction and still get the correct answer.

Worked example 1: Basic distance calculation

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Worked Example: Finding Distance Between Two Points

Question: Find the distance between $S(-2; -5)$ and $T(7; -2)$ .

Solution:

Step 1: Draw a sketch to visualise the problem.

Step 2: Assign coordinate values.

Point $S$ : $x_1 = -2$ , $y_1 = -5$
Point $T$ : $x_2 = 7$ , $y_2 = -2$

Step 3: Write the distance formula.

$d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$

Step 4: Substitute the values.

$d_{ST} = \sqrt{(-2 - 7)^2 + (-5 - (-2))^2}$
$= \sqrt{(-9)^2 + (-3)^2}$
$= \sqrt{81 + 9}$
$= \sqrt{90}$
$= 9.5 \text{ units}$

Step 5: Write the final answer.

The distance between $S$ and $T$ is $9.5$ units.

Worked example 2: Finding unknown coordinates

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Worked Example: Finding Unknown Coordinate When Distance is Given

Question: Given $RS = 13$ , $R(3; 9)$ and $S(8; y)$ , find the value of $y$ .

Solution:

Step 1: Draw a sketch showing the possible positions.

Note that there will be two possible values for $y$ because the distance formula contains $(y_1 - y_2)^2$ , which creates a quadratic equation.

Step 2: Assign coordinate values.

Point $R$ : $x_1 = 3$ , $y_1 = 9$
Point $S$ : $x_2 = 8$ , $y_2 = y$

Step 3: Write the distance formula.

$d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$

Step 4: Substitute values and solve for $y$ .

$13 = \sqrt{(3 - 8)^2 + (9 - y)^2}$
$13^2 = (-5)^2 + (9 - y)^2$
$169 = 25 + (9 - y)^2$
$144 = (9 - y)^2$
$\pm 12 = 9 - y$

This gives us: $y = 9 - 12 = -3$ or $y = 9 + 12 = 21$

Step 5: Check both values.

For $y = -3$ :

$d = \sqrt{(3-8)^2 + (9-(-3))^2} = \sqrt{25 + 144} = \sqrt{169} = 13$ ✓

For $y = 21$ :

$d = \sqrt{(3-8)^2 + (9-21)^2} = \sqrt{25 + 144} = \sqrt{169} = 13$ ✓

Step 6: Write the final answer.

Therefore, $y = -3$ or $y = 21$ .

Exam tips and techniques

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Essential Exam Strategies

Always draw a sketch first - this helps visualise the problem and makes it easier to check your answer
Be careful with negative coordinates - pay attention to signs when substituting values
Remember that distance is always positive - if you get a negative answer, check your calculations
When finding unknown coordinates, expect two possible answers - the quadratic nature of the formula often gives two solutions
Always verify your answer - substitute back into the original distance formula to check

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Key Points to Remember:

A point is written as an ordered pair $(x; y)$
Distance is the length between two points, always positive
The distance formula is $d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$
Always sketch the problem before calculating
When finding unknown coordinates, there are often two possible solutions

Distance Between Two Points (Grade 10 NSC Matric Mathematics): Revision Notes