How do you test whether three points are collinear?

Compare the slopes of two of the segments; equal slopes mean the points lie on one line, and a check against the line equation also works.

What is the difference between internal and external division?

An internal point lies between the endpoints and is a weighted average of them, while an external point lies beyond one end and its formula uses a negative ratio.

Why does a perpendicular bisector pass through the midpoint?

By definition it is perpendicular to the segment and passes through its midpoint, so the midpoint comes first and then the negative reciprocal slope.

What is the slope of the general form Ax + By + C = 0?

The slope is -A / B when B is not zero; if B is zero the line is vertical and has no slope.

S4 Maths coordinate geometry paper 2: bisectors, division points and collinearity

2026-11-10

Tutor Prep Guide · S4 Maths Coordinate Geometry

Ten items apply distance and slope to perpendicular bisectors, internal division, collinearity tests and perpendicular line equations.

Direct answers　Items 01 to 03 cover distance, a missing endpoint and slope from the general form, 04 to 06 a perpendicular line, collinearity and a bisector, 07 to 09 intercepts, internal division and a missing coordinate, and 10 a dropped negative sign.

01　Paper details (check answers afterwards)

Item	Detail
Questions	10
Suggested time	25
Difficulty mix	Foundation 3 · Intermediate 3 · Challenge 3 · Error redo 1
Chapter	Chapter: S4 Maths Coordinate Geometry
Level	S4

02　Questions (10 items)

FoundationFind the distance between (2, -1) and (7, 11).
FoundationA midpoint is (5, 2) and one end is (3, -1). Find the other end.
FoundationFind the slope of the line 4x – 2y + 6 = 0.
IntermediateFind the line through (3, 4) perpendicular to y = 2x + 1.
IntermediateAre A(1, 2), B(3, 6) and C(5, 10) collinear?
IntermediateFind the perpendicular bisector of (2, 2) and (6, 8).
ChallengeFind the x-intercept of the line 3x – 4y = 12.
ChallengeP divides the segment from A(2, 4) to B(10, 12) internally in AP : PB = 1 : 3. Find P.
ChallengeA line through (2, 3) and (4, k) has slope 4. Find k.
Error redoWhat is the slope perpendicular to a slope of 2? A student answered 1/2.

03　Answers and one-line notes

#	Answer	One-line note
1	13	The changes are 5 and 12, so the distance is the square root of 25 + 144, which is 13.
2	(7, 5)	The other end is twice the midpoint minus the known end, that is (10 – 3, 4 + 1) = (7, 5).
3	2	Rearranging gives y = 2x + 3, and the coefficient of x is the slope, so the slope is 2.
4	x + 2y = 11	The perpendicular slope is -1/2, so y – 4 = -(x – 3)/2, which rearranges to x + 2y = 11.
5	Collinear, since AB and BC both have slope 2.	AB has slope 4 / 2 = 2 and BC has slope 4 / 2 = 2; equal slopes sharing B put all three points on one line.
6	2x + 3y = 23	The midpoint is (4, 5) and the segment slope is 6 / 4 = 1.5, so the perpendicular slope is -2/3 and 2x + 3y = 23.
7	4	Putting y = 0 gives 3x = 12, so x = 4 and the intercept is (4, 0).
8	(4, 6)	The segment has four equal parts and P is one part from A: x = 2 + 8 / 4 = 4 and y = 4 + 8 / 4 = 6.
9	11	The slope is (k – 3) / (4 – 2) = 4, so k – 3 = 8 and k = 11.
10	-1/2; the answer 1/2 takes the reciprocal but drops the negative sign.	Perpendicular slopes multiply to -1, so the slope is -1 / 2 = -1/2; 1/2 would describe a line rising the other way.

04　What to do after marking

After marking, redo only the wrong items and write down which step went wrong (formula, unit, or misread number).
Repeat the same item two days later; move to the next paper only when both attempts are correct.
Parents only need the answer and note columns; there is no need to explain every item.

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