Coordinate geometry flashcards

Topic 3: Coordinate geometry

Gradient (meaning)

How much y changes for each 1 unit increase in x. Rising to the right is positive; falling to the right is negative.

Gradient of the line through (x₁, y₁) and (x₂, y₂)

m = (y₂ − y₁) ÷ (x₂ − x₁) Subtract in the same order on top and below.

Length of the line segment from (x₁, y₁) to (x₂, y₂)

√((x₂ − x₁)² + (y₂ − y₁)²)

Midpoint of (x₁, y₁) and (x₂, y₂)

((x₁ + x₂)/2, (y₁ + y₂)/2): add the coordinates and halve each.

Find the midpoint and length of the segment from (−1, −2) to (5, 6)

Midpoint: (4/2, 4/2) = (2, 2). Length: √(6² + 8²) = √100 = 10.

In y = mx + c, what do m and c stand for?

m is the gradient (the number multiplying x); c is the y-intercept, where the line crosses the y-axis (x = 0).

Find the gradient and y-intercept of 3x − 4y = 24

Rearrange: −4y = 24 − 3x, so y = (3/4)x − 6. Gradient 3/4, y-intercept −6.

Parallel lines and perpendicular lines: gradient rules

Parallel: equal gradients, m₁ = m₂. Perpendicular: m₁ × m₂ = −1, so the perpendicular gradient is −1/m (flip the fraction and change the sign).

Find the line parallel to y = 3x + 1 through (2, 5)

Gradient 3. Substitute: 5 = 3 × 2 + c, so c = −1. y = 3x − 1

Perpendicular bisector (meaning)

A line that cuts a line segment in half at 90°.

How do you find the equation of the perpendicular bisector of PQ?

1. Find the midpoint of PQ. 2. Find the gradient of PQ. 3. Take the perpendicular gradient −1/m. 4. Use y − y₁ = m(x − x₁) through the midpoint.

The lines x = 3 and y = 3: how do they differ?

x = 3 is vertical, through (3, 0); its gradient is undefined. y = 3 is horizontal, through (0, 3); its gradient is 0.