Coordinate Geometry: Straight Lines

1Summary 重點摘要

On the coordinate plane: the distance between two points uses Pythagoras (distance formula), the slope = (y₂−y₁)/(x₂−x₁) measures steepness, and in the line equation y = mx + c, m is the slope and c is the y-intercept. These three tools plus Pythagoras are the core of S3 coordinate geometry.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Recapping the coordinate systemA position on the plane is fixed by coordinates (x, y): move x along the horizontal axis, right positive, then y along the vertical, up positive. Thus (3, 2) is 3 right and 2 up from the origin (0, 0) where the axes cross. Coordinate geometry's core idea is describing shapes by algebraic equations, turning figure questions into computations. This lesson studies the straight line: every point of one line satisfies the same linear equation in x and y — the line's identity card.
2. The gradient formulaThe gradient measures steepness with m = (y₂ − y₁) ÷ (x₂ − x₁), vertical change over horizontal change. For the line through (1, 2) and (3, 6): y₂ − y₁ = 6 − 2 = 4 and x₂ − x₁ = 3 − 1 = 2, giving a gradient of 4 ÷ 2 = 2. The sign gives the direction: positive climbs, negative descends, zero is level. When substituting, subtract the two points' x and y in the same order: if y takes 'second minus first', x must too — inconsistent order flips the sign and misreads the direction.
3. The line equation y = mx + cThe standard form is y = mx + c: m is the gradient and c the y-intercept, the height where the line crosses the y-axis. For y = 2x + 1, m = 2 and c = 1, crossing at (0, 1). The form's power: knowing m and c is knowing the whole line; conversely, reading a line's tilt and crossing gives back its equation. To test whether a point lies on the line, substitute its coordinates: a true statement means on the line, a false one means off it — a one-step verification.
4. Writing the equation from two pointsGiven the line through (0, 1) and (2, 5): first the gradient m = (5 − 1) ÷ (2 − 0) = 4 ÷ 2 = 2; second, (0, 1) is itself the y-axis crossing, so c = 1, and the equation is y = 2x + 1. When neither point is the intercept, as with (1, 2) and (3, 6), find m = 2, then set the equation as y = 2x + c and substitute (1, 2): 2 = 2 + c, so c = 0 and the line is y = 2x. Verify with the other point before finishing, and the equation is safe.
5. Horizontal and vertical linesOn a horizontal line every point shares one y, so the equation reads y = 3 and the gradient is 0; on a vertical line every point shares one x, so the equation reads x = 2 and the gradient is undefined, because the horizontal run is 0 and division by zero is impossible. The memory rule: y equals a number gives a horizontal line; x equals a number gives a vertical one. The standard error is swapping the two. Both special lines appear often in figure boundaries and simultaneous equations, so recognise them instantly.
6. Distance between two points and combined useThe distance between A(x₁, y₁) and B(x₂, y₂) is AB = √[(x₂ − x₁)² + (y₂ − y₁)²], coming from Pythagoras: the horizontal and vertical differences are the two legs of a right triangle and AB is the hypotenuse. For A(1, 2) and B(4, 8): differences 3 and 6 give AB = √(9 + 36) = √45 = 3√5 ≈ 6.7. Combined coordinate-geometry questions often test slope, equation and length on the same line; proceed in the order slope, intercept, distance and the full marks follow steadily.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Coordinates: (x, y)
• Slope = (y₂−y₁)/(x₂−x₁)
• Line equation: y = mx + c (m slope, c y-intercept)
• y = number = horizontal, x = number = vertical
• (0, c) is the y-intercept point