Section B — Chapters 23–39

Co-ordinate Geometry II: Midpoint, Slope, and Distance

Junior Cycle — 1st Year

✓By the end of this lesson students will be able to calculate the midpoint of a line segment given the co-ordinates of its endpoints.
✓By the end of this lesson students will be able to calculate the slope (gradient) of a line given the co-ordinates of two points on the line.
✓By the end of this lesson students will be able to calculate the distance between two points given their co-ordinates.
✓By the end of this lesson students will be able to apply these co-ordinate geometry formulae to solve simple problems.

Key concepts

Midpoint of a Line Segment

The midpoint of a line segment is the point that lies exactly halfway between its two endpoints. To find the midpoint, we average the x-co-ordinates and average the y-co-ordinates of the two given points.

Given two points (x₁, y₁) and (x₂, y₂), the midpoint M is given by M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

Slope of a Line

The slope of a line, often denoted by 'm', measures its steepness or gradient. It tells us how much the line rises or falls vertically for every unit it moves horizontally. A positive slope means the line goes up from left to right, a negative slope means it goes down, a zero slope is a horizontal line, and an undefined slope is a vertical line.

Given two points (x₁, y₁) and (x₂, y₂), the slope m is given by m = (y₂ - y₁) / (x₂ - x₁) or m = (change in y) / (change in x)

Distance Between Two Points

The distance between two points is the length of the straight line segment connecting them. This formula is derived directly from Pythagoras' theorem, considering the horizontal and vertical distances between the points as the legs of a right-angled triangle.

Given two points (x₁, y₁) and (x₂, y₂), the distance d is given by d = √((x₂ - x₁)² + (y₂ - y₁)²)

Key facts to remember

1The midpoint formula is M = ((x₁ + x₂)/2, (y₁ + y₂)/2).
2The slope formula is m = (y₂ - y₁) / (x₂ - x₁).
3The distance formula is d = √((x₂ - x₁)² + (y₂ - y₁)²).
4A horizontal line has a slope of 0 (y₁ = y₂).
5A vertical line has an undefined slope (x₁ = x₂).
6Distance is always a positive value.

Worked examples

Example 1

Given the points A(2, 3) and B(8, 11), find:\n(a) The midpoint of the line segment AB.\n(b) The slope of the line AB.\n(c) The distance between points A and B.

ILet (x₁, y₁) = (2, 3) and (x₂, y₂) = (8, 11).

II(a) Midpoint M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

IIIM = ((2 + 8)/2, (3 + 11)/2)

IVM = (10/2, 14/2)

VM = (5, 7)

VI(b) Slope m = (y₂ - y₁) / (x₂ - x₁)

VIIm = (11 - 3) / (8 - 2)

VIIIm = 8 / 6

9m = 4/3

10(c) Distance d = √((x₂ - x₁)² + (y₂ - y₁)²)

11d = √((8 - 2)² + (11 - 3)²)

12d = √((6)² + (8)²)

13d = √(36 + 64)

14d = √(100)

15d = 10

Answer

(a) Midpoint = (5, 7)\n(b) Slope = 4/3\n(c) Distance = 10 units

Remember to simplify fractions for the slope.

Example 2

Find the slope of the line passing through P(-3, 5) and Q(7, -1).

ILet (x₁, y₁) = (-3, 5) and (x₂, y₂) = (7, -1).

IISlope m = (y₂ - y₁) / (x₂ - x₁)

IIIm = (-1 - 5) / (7 - (-3))

IVm = -6 / (7 + 3)

Vm = -6 / 10

VIm = -3/5

Answer

The slope of the line is -3/5.

Be very careful with negative signs when substituting values into the formula.

Example 3

A line segment has one endpoint at C(1, -4) and its midpoint is M(4, 0). Find the co-ordinates of the other endpoint, D(x, y).

ILet C = (x₁, y₁) = (1, -4) and M = (x_m, y_m) = (4, 0). Let D = (x₂, y₂).

IIWe know the midpoint formula: (x_m, y_m) = ((x₁ + x₂)/2, (y₁ + y₂)/2)

IIISubstitute the known values:

IV(4, 0) = ((1 + x₂)/2, (-4 + y₂)/2)

VEquate the x-co-ordinates: 4 = (1 + x₂)/2

VIMultiply both sides by 2: 8 = 1 + x₂

VIISubtract 1 from both sides: x₂ = 7

VIIIEquate the y-co-ordinates: 0 = (-4 + y₂)/2

9Multiply both sides by 2: 0 = -4 + y₂

10Add 4 to both sides: y₂ = 4

11So, the co-ordinates of D are (7, 4).

Answer

The co-ordinates of the other endpoint D are (7, 4).

This problem requires working backwards from the midpoint formula. Break it into two separate equations for x and y.

Common mistakes

✗Mixing up the x and y co-ordinates in the formulae.
✗Incorrectly handling negative signs, especially in the slope and distance formulae.
✗Forgetting to divide by 2 for both x and y when finding the midpoint.
✗Making calculation errors with squares and square roots in the distance formula.
✗Not simplifying the fraction for the slope to its lowest terms.

Exam tips

★Always write down the formula you are using before substituting values. This helps you remember it and can earn you marks even if you make a calculation error.
★Label your points (x₁, y₁) and (x₂, y₂) clearly to avoid confusion, especially with negative co-ordinates.
★Show all your steps clearly. This makes your work easy to follow and helps you spot any mistakes.
★Use a calculator carefully for the distance formula, especially with square roots. Round your answer to the specified number of decimal places if required.

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