Mathematics · Lesson 19

Coordinate geometry of straight lines

JAMBWAECNECO

What the syllabus asks

JAMB: Coordinate geometry: midpoint and gradient of a line segment, distance between two points, parallel and perpendicular lines, equations of straight lines

WAEC: Coordinate geometry of straight lines: the Cartesian plane, distance between two points, midpoint of a line segment, gradient of a line, equation of a straight line (gradient–intercept, point–gradient and two-point forms), intercepts, parallel and perpendicular lines, intersection of two lines

NECO: Coordinate geometry of straight lines: the Cartesian plane, distance between two points, midpoint of a line segment, gradient of a line, equation of a straight line (gradient–intercept, point–gradient and two-point forms), intercepts, parallel and perpendicular lines, intersection of two lines

What you will learn

Plot points and read coordinates on the Cartesian plane.
Calculate the distance between two points and the midpoint of a segment.
Find the gradient of a line from two points or from its equation, and relate it to the angle of inclination.
Write the equation of a line in the forms y = mx + c, y − y₁ = m(x − x₁) and ax + by + c = 0.
Find intercepts and use them to sketch a line.
Use gradients to decide whether lines are parallel or perpendicular, and find equations of such lines.
Find the point where two lines meet, test points for collinearity, and find the area of a triangle from its vertices.

Key points

A point is (x, y): across first, then up or down.
Distance = √[(x₂ − x₁)² + (y₂ − y₁)²].
Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2).
Gradient = (y₂ − y₁)/(x₂ − x₁) = tan θ; horizontal lines have m = 0 and vertical lines have undefined gradient.
y = mx + c: m is the gradient, c the y-intercept; for ax + by + c = 0, m = −a/b.
Line through (x₁, y₁) with gradient m: y − y₁ = m(x − x₁).
Parallel lines: equal gradients.
Perpendicular lines: m₁m₂ = −1.
Intersections come from solving the equations simultaneously.
Collinear points share a gradient; area of a triangle = ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|.

Start here: The Cartesian plane

Imagine giving a friend directions in a new estate in Port Harcourt: “walk 3 streets east, then 2 streets north.” Two numbers fix the spot exactly. The Cartesian plane does the same for points on paper, and coordinate geometry lets us answer questions about lengths, slopes and shapes with algebra instead of a ruler.

The horizontal axis is the x-axis and the vertical axis is the y-axis. They meet at the origin O(0, 0). A point is written (x, y): the x-coordinate first (across), then the y-coordinate (up or down). The axes divide the plane into four quadrants: (+, +) top right, (−, +) top left, (−, −) bottom left and (+, −) bottom right.

In the diagram each grid unit is 20 pixels: A(3, 2) is 3 right and 2 up; B(−3, −2) is 3 left and 2 down; C(−2, 3) is 2 left and 3 up.

Worked example 19.1

In which quadrant does each point lie: P(4, −1), Q(−5, 6), R(0, 3)?

P: x positive, y negative, so the 4th quadrant (bottom right).
Q: x negative, y positive, so the 2nd quadrant (top left).
R: x = 0, so it lies on the y-axis itself, not in any quadrant.

P: 4th; Q: 2nd; R: on the y-axis

⚠️ Common mistake: plotting (2, 5) as 2 up and 5 across. The x-coordinate is always first. Remember “along the corridor, then up the stairs”.

Try it yourself. Write down the coordinates of a point 4 units left of the origin on the x-axis.

Show solution

(−4, 0). Points on the x-axis have y = 0.

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Practice questions

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

Show worked answer

√(36 + 64) = 10 units.

2. Find the midpoint of (−1, 7) and (5, −3).

Show worked answer

(2, 2).

3. Find the gradient and y-intercept of 5x − 2y = 8.

Show worked answer

y = 2.5x − 4: m = 5/2, c = −4.

4. Find the equation of the line through (−2, 3) and (4, −9).

Show worked answer

m = −2; y = −2x − 1.

5. Find the equation of the line through (1, 4) perpendicular to 3x + y = 7.

Show worked answer

Given m = −3, so m = ⅓: y − 4 = ⅓(x − 1), x − 3y + 11 = 0.

Key words

Cartesian plane: A flat surface with a horizontal x-axis and a vertical y-axis meeting at the origin.
Origin: The point (0, 0) where the axes cross.
Coordinates: The ordered pair (x, y) that fixes a point’s position.
Line segment: The part of a line between two end points.
Midpoint: The point exactly halfway along a line segment.
Gradient (slope): The steepness of a line: vertical change ÷ horizontal change.
y-intercept: The y-value where a line cuts the y-axis.
x-intercept: The x-value where a line cuts the x-axis.
Angle of inclination: The angle a line makes with the positive x-axis, measured anticlockwise.
Parallel lines: Lines with the same gradient that never meet.
Perpendicular lines: Lines that meet at right angles.
Negative reciprocal: The number −1/m; the gradient perpendicular to gradient m.

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