A line through A(−4, 4) and B(2, k) has slope −3/2. k = □

−5. (k − (4))/(2 − (−4)) = −3/2, so k − (4) = −3/2 × 6 = −9 and k = −5.

P lies on AB extended, with A(−6, −2), B(2, 6) and AP : PB = 5 : 1. P = (□, □). Fill in each blank

4, 8. External division: P = ((5 × (2) − 1 × (−6))/4, (5 × (6) − 1 × (−2))/4) = (16/4, 32/4) = (4, 8).

Coordinates and trigonometry · Ages 14–15

From “The distance formula” to “External division”, step by step, each skill with its methods, worked examples and answers.

The distance formulaThe run and the rise are the legs, AB the hypotenuse
MethodA(1, 2), B(4, 6): 3² + 4² = 5
Another way: Draw the right triangle3 squares across, 4 up: the hypotenuse is 5
The mid-point formulaThe mid-point is the average of the ends
MethodA(2, 7), B(8, −1): M = (5, 3)
Another way: Half the way from A2 + 6 ÷ 2 = 5, 7 − 8 ÷ 2 = 3
The section formula (internal division)s times x₁ and r times x₂, over r + s
MethodAP : PB = 1 : 2: P = (2 × 1 + 1 × 73, 2 × 2 + 1 × 113) = (3, 5)
Another way: Walk r/(r + s) of the way1 + 6 × 13 = 3, 2 + 9 × 13 = 5
SlopeSlope = rise ÷ run, both subtracted the same way
MethodA(1, 2), B(5, 8): 8 − 25 − 1 = 32
Another way: Count squares4 right, 6 up: 64 = 32
Using slopesIn a line: equal slopes; an unknown: an equation in the slope
Methodk − 36 − 2 = 12, k = 5
Another way: Walk it4 × 12 = 2, 3 + 2 = 5
Slopes of parallel and perpendicular linesParallel: equal slopes; perpendicular: product −1
Method23 × (−32) = −1
Another way: Turn it over, change the sign23 → 32 → −32
Proofs with coordinatesRight angle: product −1 or the converse; parallel: equal slopes
Method12 × (−2) = −1, so AB ⊥ BC
Another way: By the lengthsAB² + BC² = 20 + 20 = 40 = AC²
External divisionExternal: r x₂ − s x₁r − s
MethodAP : PB = 3 : 1: x = 3 × 3 − 1 × 12 = 4
Another way: Walk itAB is 2 parts; from B one more: 3 + 2 ÷ 2 = 4

From different skills in this topic. Have a go first, then open each one for the answer and the working.

Example 1The distance formula →
In the picture, from A to B is units across and units up or down, so AB = .
Fill in each blank
Show the answer and working
Answer: 3, 4, 5
3 across and 4 up or down are the legs of a right triangle and AB its hypotenuse: 3² + 4² = 5.
Example 2Using slopes →
A line through A(−4, 4) and B(2, k) has slope −32. k =
Show the answer and working
Answer: −5
k − (4)2 − (−4) = −32, so k − (4) = −32 × 6 = −9 and k = −5.
Example 3External division →
P lies on AB extended, with A(−6, −2), B(2, 6) and AP : PB = 5 : 1. P = (, ).
Fill in each blank
Show the answer and working
Answer: 4, 8
External division: P = (5 × (2) − 1 × (−6)4, 5 × (6) − 1 × (−2)4) = (164, 324) = (4, 8).

Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.

Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
Practice paper: Set out in parts like a school paper, marked, with answers and full working