- 1. The Distance Between Two Points
Derive the distance formula from Pythagoras and use it on any pair of points, in surd form or correct to a given number of decimal places.
- 2. Using Distance: What Shape Is It?
Turn computed lengths into proof: classify triangles, test for a right angle with the converse of Pythagoras, and show that a point is equidistant from two others.
- 3. The Midpoint of a Line Segment
Find the midpoint by averaging coordinates, recover a missing endpoint from a given midpoint, and use the shared midpoint of a parallelogram's diagonals.
- 4. Gradient: Steepness as a Number
Compute the gradient of a line from two points, interpret its sign and size, handle the horizontal and vertical cases, and prove three points collinear.
- 5. y = mx + c: Gradient and Intercepts
Read the gradient and y-intercept from an equation, sketch a line without a table, find both intercepts, and rearrange a general-form equation into gradient form.
- 6. Parallel and Perpendicular Lines
Use m1 = m2 for parallel lines and m1m2 = -1 for perpendicular lines, including lines given in general form, and apply both conditions to find unknowns and classify quadrilaterals.
- 7. Finding the Equation of a Line From Data
Build the equation of a line from a gradient and a point or from two points, using the point-gradient form, and present it in general or intercept form.
- 8. The Perpendicular Bisector
Describe the perpendicular bisector as a locus, build its equation from a midpoint and a negative reciprocal, verify it, and derive the same line from |PA| = |PB|.
- 9. The Angle Between Two Intersecting Lines
Convert between a gradient and an angle of slope, find the acute angle between two lines from their gradients, and describe a crossing by its point and its angle.
- 10. The Triangle Problem: Putting It All Together
Choose the right tool from the wording of a question: medians, altitudes, feet of perpendiculars, bisectors and angles on one triangle - and see why a curve needs a gradient at a point.
