SECONDARY MATHEMATICS · COORDINATE GEOMETRY REASONING
Gradient turns direction into a number. That lets coordinate geometry test parallelism and perpendicularity without trusting a sketch.
Parallel non-vertical lines have equal gradients
A line through A(1,2) and B(4,8) has gradient (8−2)/(4−1)=2. A line through C(0,5) and D(3,11) also has gradient 2. The lines have the same direction and, because they are distinct, are parallel.
Equal gradients alone require care if the two descriptions could refer to the same line. “Parallel distinct lines” and “coincident line” should not be confused.
Perpendicular non-vertical/non-horizontal lines
For two non-vertical lines with nonzero gradients m₁ and m₂, perpendicularity corresponds to m₁m₂=−1. Thus a line with gradient 2 is perpendicular to a line with gradient −1/2.
This negative-reciprocal relationship encodes a right-angle turn in coordinate form.
Horizontal and vertical lines are a special case
A horizontal line has gradient 0. A vertical line has undefined gradient. They are perpendicular to each other, so the product-of-gradients test cannot be applied mechanically because one gradient is undefined.
Prove a right angle from coordinates
Let A(1,1), B(5,3) and C(6,1). Gradient AB=(3−1)/(5−1)=1/2. Gradient BC=(1−3)/(6−5)=−2. Their product is −1, so AB is perpendicular to BC and angle ABC is 90°.
Prove opposite sides parallel
For quadrilateral A(0,0), B(4,2), C(7,6), D(3,4): gradient AB=1/2 and gradient DC=(6−4)/(7−3)=1/2. Gradient BC=(6−2)/(7−4)=4/3 and gradient AD=(4−0)/(3−0)=4/3. Both pairs of opposite sides are parallel, supporting the conclusion that ABCD is a parallelogram.
Do not use gradient when a simpler coordinate fact is enough
If two points share the same y-coordinate, the segment is horizontal. If two share the same x-coordinate, it is vertical. Coordinate reasoning should use the simplest justified structure available.
Worked checks
A. Gradient 3 and gradient 3: parallel if the lines are distinct.
B. Gradient 4 and gradient −1/4: perpendicular.
C. Gradient −2 and gradient 1/2: perpendicular.
D. Gradient 0 and a vertical line: perpendicular, handled as the horizontal–vertical special case.
Continue
Build from Coordinates as Geometry and What a Gradient Really Measures, then continue into midpoint, distance and coordinate proof. Return to the Secondary Mathematics Master Index.