This mini-lesson covers Coordinate Geometry for Cambridge IGCSE Mathematics (0580): gradient, midpoint, length of a line segment, the equation of a straight line, and parallel & perpendicular lines (Extended content included).
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Gradient
Gradient of a line
The gradient measures steepness: rise over run between two points.
m = (y₂ − y₁) ÷ (x₂ − x₁)change in y divided by change in x
Worked example
From (1, 2) to (4, 11): m = (11 − 2)/(4 − 1) = 9/3 = 3.
Sign matters: a line going down left-to-right has a negative gradient.
Calculate
Your turn
1Find the gradient of the line joining (2, 3) and (6, 15).
Hint: (15 − 3) ÷ (6 − 2) = 12 ÷ 4.
Midpoint
Midpoint of a segment
The midpoint is the average of the coordinates.
midpoint = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )average the x's, average the y's
Worked example
Midpoint of (2, 4) and (8, 10) = ((2+8)/2, (4+10)/2) = (5, 7).
Calculate
Your turn
2Find the x-coordinate of the midpoint of (3, 5) and (11, 9).
Hint: (3 + 11) ÷ 2.
Length
Length of a line segment
Use Pythagoras on the horizontal and vertical differences.
length = √[(x₂ − x₁)² + (y₂ − y₁)²]the straight-line distance between two points
Worked example
From (0, 0) to (3, 4): √(3² + 4²) = √25 = 5.
Calculate
Your turn
3Find the length of the segment from (1, 2) to (7, 10).
Hint: differences are 6 and 8; √(6² + 8²) = √100.
Equation of a line
Equation of a straight line
The straight-line equation is y = mx + c, where m is the gradient and c is the y-intercept.
y = mx + cm = gradient · c = value of y where the line crosses the y-axis
Worked example
Gradient 2 through (0, 3): y = 2x + 3, so c = 3.
Calculate
Your turn
4A line has gradient 4 and passes through (0, −5). Its equation is y = 4x + c. Type c.
Hint: c is the y-value where x = 0, which is −5.
Quick check
Quick check
?What is the gradient of the line y = 7 − 3x?
Parallel & perpendicular
Parallel & perpendicular lines
Parallel lines have the same gradient. Perpendicular gradients multiply to −1 — take the negative reciprocal.
m₁ × m₂ = −1if m = 2, the perpendicular gradient is −½
Perpendicular ≠ just negative: the perpendicular to gradient 3 is −⅓ (flip and change sign), not −3.
Calculate
Your turn
5A line has gradient 5. A line perpendicular to it has gradient −1/k. Type the value of k (so the gradient is −1/5).
Hint: perpendicular gradient = −1 ÷ 5 = −1/5, so k = 5.
Quick check
Quick check
?A line has gradient ½. What is the gradient of a line perpendicular to it?
Match game
Match line to gradient
Tap a line equation, then its gradient.
Calculate
Your turn
6Line A is y = 3x + 1. Line B is parallel to A and passes through (0, 7). Its equation is y = mx + c. Type the gradient m.
Hint: parallel lines share the same gradient as line A.
Sort it
Parallel to y = 2x?
Parallel lines have gradient 2. Sort each line: is it parallel to y = 2x? Tap it, then its box.
∥ Parallel (m = 2)
✗ Not parallel
Recap
Key points
Gradient: (y₂ − y₁) ÷ (x₂ − x₁).
Midpoint: average the x's and the y's.
Length: √[(Δx)² + (Δy)²] (Pythagoras).
Line: y = mx + c, m = gradient, c = y-intercept.
Parallel: same gradient. Perpendicular: negative reciprocal (m₁m₂ = −1).
You've covered the Coordinate Geometry essentials for Cambridge IGCSE Maths (0580). Press Finish to see your score.
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