Analytical Geometry
Distance, midpoint, gradient on the Cartesian plane
Distance Between Two Points
The distance between points A(x1, y1) and B(x2, y2) is:
d = sqrt[(x2 − x1)² + (y2 − y1)²]
This is based on the Pythagorean theorem applied to the coordinate plane.
Example
Distance Example
Find the distance between A(1, 2) and B(4, 6).
d = sqrt[(4 − 1)² + (6 − 2)²]
d = sqrt[9 + 16]
d = sqrt[25]
d = 5 units
Midpoint and Gradient
Midpoint of A(x1, y1) and B(x2, y2):
M = ((x1 + x2)/2, (y1 + y2)/2)
Gradient (slope) of line AB:
m = (y2 − y1) / (x2 − x1)
Parallel lines have equal gradients. Perpendicular lines: m1 × m2 = −1.
Example
Midpoint and Gradient Example
A(2, 3) and B(8, 7)
Midpoint M = ((2+8)/2, (3+7)/2) = (5, 5)
Gradient m = (7−3)/(8−2) = 4/6 = 2/3
A line perpendicular to AB has gradient: −3/2 (negative reciprocal).
Note
Remember
Distance: use the formula or draw a right triangle on the grid. Midpoint: average the x-values and average the y-values. Gradient: rise over run = (change in y)/(change in x). Vertical lines have undefined gradient.
Key Vocabulary
CoordinateAn ordered pair (x, y) that gives a point's position
Distance formulad = sqrt[(x2-x1)² + (y2-y1)²]
MidpointThe point exactly halfway between two points
GradientThe steepness of a line (rise over run)
PerpendicularLines that meet at a right angle (90°)
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