Mathematics·Coordinate geometry · NSSCO 8.1
Coordinate geometry: gradient, midpoint, distance & line equations
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Give me any two points on the grid, and I can tell you almost everything about the line that joins them. In this lesson we take a single pair of points, A and B, and squeeze four beautiful facts out of them: the gradient (how steep the join is), the midpoint (the exact middle), the distance (how long the join is, using Pythagoras), and the equation of the straight line itself in the form y = mx + c. Then we meet the tidy form ax + by + c = 0, and finish with the two rules examiners love — parallel lines share a gradient, and perpendicular lines have gradients that multiply to make −1. Every step is written on the board line by line, so you can copy it into your own book as we go.
What you'll learn in this lesson
By the end you should be able to (NSSCO Mathematics 8.1):
- Calculate the gradient of the line segment joining two points as (y₂ − y₁) ÷ (x₂ − x₁)
- Find the coordinates of the midpoint of a segment by averaging the x-coordinates and the y-coordinates
- Calculate the distance (length) between two points using Pythagoras' theorem
- Find the equation of a straight line y = mx + c given a point and the gradient, or given two points on it
- Interpret and use equations of the form ax + by + c = 0
- Know that parallel lines have equal gradients
- Know that the gradients of perpendicular lines multiply to give −1 (they are negative reciprocals)
- Apply coordinate geometry to decide facts about quadrilaterals (e.g. opposite sides parallel)
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