Mid-Point of a Line Revision Notes for Grade 10 NSC Matric Mathematics

Mid-Point of a Line

What is a mid-point?

The mid-point of a line segment is the point that lies exactly halfway between two given endpoints. Think of it as the "middle" point that divides a line segment into two equal parts. On a coordinate plane, we can find this middle point by calculating the average of the x-coordinates and the average of the y-coordinates of the two endpoints.

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When you have two points on a coordinate plane, the mid-point will always lie on the straight line connecting these points, positioned at equal distances from both endpoints. This geometric property makes the mid-point formula a powerful tool for coordinate geometry problems.

The mid-point formula

To find the coordinates of the mid-point M(x; y) between any two points A(x₁; y₁) and B(x₂; y₂), we use the following formula:

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The Mid-Point Formula:

$M(x; y) = \left(\frac{x_1 + x_2}{2}; \frac{y_1 + y_2}{2}\right)$

This can be written as two separate equations:

$x = \frac{x_1 + x_2}{2}$
$y = \frac{y_1 + y_2}{2}$

The formula works by taking the arithmetic mean (average) of the corresponding coordinates. This averaging process ensures that the resulting point is positioned exactly halfway between the two original points.

Step-by-step method for calculating mid-points

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Following a systematic approach will help you solve mid-point problems accurately and avoid common errors. Always work through these steps in order to ensure you don't miss any important details.

Follow these clear steps when solving mid-point problems:

Draw a sketch - Always plot the given points on a coordinate plane to visualise the problem
Identify the coordinates - Label the points clearly as (x₁; y₁) and (x₂; y₂)
Write down the formula - State the mid-point formula clearly
Substitute the values - Replace the variables with the actual coordinate values
Calculate and simplify - Work through the arithmetic carefully
Write the final answer - Express your answer as coordinate pair (x; y)
Check your answer - Verify that your result makes sense by looking at your sketch

Worked examples

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Worked Example 1: Finding a mid-point between two points

Question: Calculate the coordinates of the mid-point F(x; y) of the line between point E(2; 1) and point G(-2; -2).

Solution:

Step 1: Draw a sketch (shown above)

Step 2: Assign values to (x₁; y₁) and (x₂; y₂)

x₁ = -2, y₁ = -2, x₂ = 2, y₂ = 1

Step 3: Write down the mid-point formula

$F(x; y) = \left(\frac{x_1 + x_2}{2}; \frac{y_1 + y_2}{2}\right)$

Step 4: Substitute values into the formula

$x = \frac{x_1 + x_2}{2} = \frac{-2 + 2}{2} = \frac{0}{2} = 0$
$y = \frac{y_1 + y_2}{2} = \frac{-2 + 1}{2} = \frac{-1}{2}$

Step 5: Write the answer

The mid-point is at F(0; -½).

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Worked Example 2: Finding a mid-point with different coordinates

Question: Find the mid-point of line AB, given A(6; 2) and B(-5; -1).

Solution:

Step 1: Draw a sketch (shown above)

Step 2: Assign values to (x₁; y₁) and (x₂; y₂)

Let the mid-point be M(x; y)
x₁ = 6, y₁ = 2, x₂ = -5, y₂ = -1

Step 3: Write down the mid-point formula

$M(x; y) = \left(\frac{x_1 + x_2}{2}; \frac{y_1 + y_2}{2}\right)$

Step 4: Substitute values and simplify

$M(x; y) = \left(\frac{6 + (-5)}{2}; \frac{2 + (-1)}{2}\right) = \left(\frac{1}{2}; \frac{1}{2}\right)$

Step 5: Write the final answer

M(½; ½) is the mid-point of line AB.

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Worked Example 3: Finding an unknown endpoint

Question: The line joining C(-2; 4) and D(x; y) has the mid-point M(1; -3). Find point D.

Solution:

Step 1: Draw a sketch (shown above)

Step 2: Assign values to (x₁; y₁) and (x₂; y₂)

Let the coordinates of C be (x₁; y₁) and the coordinates of D be (x₂; y₂).

x₁ = -2, y₁ = 4, x₂ = x, y₂ = y

Step 3: Write down the mid-point formula

$M(x; y) = \left(\frac{x_1 + x_2}{2}; \frac{y_1 + y_2}{2}\right)$

Step 4: Substitute values and solve for x₂ and y₂

For the x-coordinate:

$1 = \frac{-2 + x_2}{2}$

$1 \times 2 = -2 + x_2$

$2 = -2 + x_2$

$x_2 = 2 + 2 = 4$

For the y-coordinate:

$-3 = \frac{4 + y_2}{2}$

$-3 \times 2 = 4 + y_2$

$-6 = 4 + y_2$

$y_2 = -6 - 4 = -10$

Step 5: Write the final answer The coordinates of point D are (4; -10).

Applications of the mid-point formula

The mid-point formula is particularly useful in:

Finding the centre of line segments in geometric constructions
Solving parallelogram problems where diagonals bisect each other
Coordinate geometry proofs involving symmetry and balance
Real-world applications such as finding the halfway point between two locations

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When working with parallelograms, remember that the diagonals bisect each other, meaning they share the same mid-point. This property can help you find unknown coordinates of vertices and is a powerful tool for solving complex coordinate geometry problems.

Key exam tips

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Essential Exam Strategies:

Always draw a sketch first - this helps you visualise the problem and check if your answer is reasonable
Be careful with negative signs - pay close attention when adding negative coordinates
Show all your working - write down each step clearly for full marks
Check your final answer - ensure your calculated mid-point lies between the two given points
Use brackets correctly - when substituting negative values, use brackets to avoid sign errors
Simplify fractions - always express your final coordinates in simplest form

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Key Points to Remember:

The mid-point formula averages the x-coordinates and y-coordinates: $M = \left(\frac{x_1 + x_2}{2}; \frac{y_1 + y_2}{2}\right)$
The mid-point always lies exactly halfway between the two given points on the line segment
Drawing a sketch is essential for visualising the problem and checking your answer makes sense
When finding unknown endpoints, substitute the known mid-point coordinates into the formula and solve the resulting equations
In parallelograms, the diagonals bisect each other, so they share the same mid-point - use this property to solve problems involving quadrilaterals

Mid-Point of a Line (Grade 10 NSC Matric Mathematics): Revision Notes