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Straight lines, parallel and perpendicular
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Gradient tells you how steep a line is. With one point and a gradient you can write the equation of any straight line.
Gradient
m = (y2 โ y1)(x2 โ x1)
(2, 3) to (6, 11): m = 84 = 2
(2, 3) to (6, 11): m = 84 = 2
Equation of a line
y โ y1 = m(x โ x1)
Through (1, 4) with gradient 3: y โ 4 = 3(x โ 1), so y = 3x + 1.
Rearranging ax + by + c = 0 into y = mx + c shows the gradient.
Through (1, 4) with gradient 3: y โ 4 = 3(x โ 1), so y = 3x + 1.
Rearranging ax + by + c = 0 into y = mx + c shows the gradient.
Parallel and perpendicular
Parallel lines have equal gradients: m1 = m2
Perpendicular lines: m1 ร m2 = โ1, so m2 = โ1m1
2x + 3y = 6 has gradient โ23. A perpendicular line has gradient 32.
Perpendicular lines: m1 ร m2 = โ1, so m2 = โ1m1
2x + 3y = 6 has gradient โ23. A perpendicular line has gradient 32.
Find the perpendicular bisector of A(1, 2) and B(5, 6).
- Midpoint (3, 4)
- Gradient AB = 44 = 1, so perpendicular gradient โ1
- y โ 4 = โ1(x โ 3)
Answer: y = โx + 7
m = (y2 โ y1)(x2 โ x1). y โ y1 = m(x โ x1). Parallel: same m. Perpendicular: m1m2 = โ1.
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