Cambridge IGCSE Additional Mathematics Notes 7: Straight-Line Graphs
Cambridge IGCSE Additional Mathematics notes on line equations, perpendicular bisectors and transformed straight-line relationships.
Cambridge IGCSE Additional Mathematics 0606 Topic 7 covers straight-line equations, parallel and perpendicular conditions, midpoint, length and perpendicular bisectors. It also owns transformation of nonlinear relationships to and from straight-line form so that gradients and intercepts determine unknown constants.
1. Gradient and line equations
For two points, gradient is change in y divided by change in x, using the same subtraction order in numerator and denominator. The form y = mx + c reveals gradient m and y-intercept c. The point-gradient form y - y₁ = m(x - x₁) is efficient when a point and gradient are known.
Other valid forms include ax + by = c and vertical lines x = k. A vertical line has undefined gradient, not zero. A horizontal line has gradient zero and equation y = k.
After forming an equation, substitute every given point and verify its gradient. Simplify into the form requested.
2. Parallel and perpendicular lines
Distinct parallel non-vertical lines have equal gradients. Perpendicular non-vertical, non-horizontal lines have gradients whose product is -1, so one is the negative reciprocal of the other.
Vertical and horizontal lines form the special perpendicular pair. Do not attempt to calculate a negative reciprocal of an undefined gradient.
Geometric conditions must be translated before forming the line. “Parallel to 3x - 2y = 7” first requires rearranging the given line or otherwise identifying its gradient.
3. Midpoint and length
The midpoint of endpoints (x₁, y₁) and (x₂, y₂) averages matching coordinates. The length is the square root of the sum of squared coordinate differences.
Keep an exact square root if later algebra uses the value or exact form is requested. Coordinate differences may be negative, but their squares make length non-negative.
These results help classify shapes, locate centres and construct perpendicular bisectors.
4. Perpendicular bisectors
A perpendicular bisector passes through a segment's midpoint and is perpendicular to the segment. The complete method is:
- find the segment gradient;
- find the perpendicular gradient;
- find the midpoint;
- form the line through that midpoint;
- verify both conditions.
If the original segment is vertical, its perpendicular bisector is horizontal through the midpoint. If it is horizontal, the bisector is vertical.
5. Why transform a relationship
A nonlinear relationship may become Y = mX + c after defining transformed variables X and Y. A straight-line plot then reveals constants through gradient and intercept.
