Chapter 4 · Geometry and Equations · Concept 1
4-6 Symmetric Points
The last lesson of the concept, and it is the midpoint formula wearing a different name. Two points are symmetric about a third exactly when that third one sits halfway between them.
Guiding question
If you know the middle and one end, where is the other end?
Learning outcome. Find the point symmetric to a given point about another point.
Symmetric about a point means midpoint
B and C are symmetric about point A if and only if A is the midpoint of BC.
So the method is two lines.
- 1. Draw it, so you know roughly where the answer belongs.
- 2. Let the point you want be (s, t), write the midpoint of the two in terms of s and t, and set it equal to the centre.
That gives two small equations, one for each coordinate, and each has a single unknown in it.
A one-line shortcut, once you believe it.
Going from A to P and then the same distance again lands you on B, so B = 2P − A. It is the same statement rearranged, and it is quick — but do the midpoint version until it feels obvious, because the shortcut is easy to write backwards as 2A − P.
Whatever it takes to get from A to the centre, do it once more. Wherever you land is the symmetric point.
Find B, the point symmetric to A(−3, 2) about P(2, 6).
- 1
Sketch the three points in a row: A, then P, then B further on.
A(−3, 2) → P(2, 6) → B(?, ?)P is in the MIDDLE, so B must be beyond it — up and to the right of P, not between the two.
Work with a partner. One uses the midpoint equations, the other uses B = 2P − A. You should land on the same point — and if not, the midpoint version is the one to trust.
Check every one the same way.
Average your answer with the point you started from. You must land exactly on the centre. It takes five seconds and it catches every arithmetic slip — for question 2, (7 + (−13))/2 = −3 and (4 + 6)/2 = 5 ✓
The points B(m, 5) and C(4, n) are symmetric about A(1, −1). Find m + n.
If you change the ratio a segment is divided in, how does the point move? And why is thinking in ratios and reference distances so useful outside maths?
Think about what a ratio measures that a distance does not.
Answering with the midpoint.
You are given the middle and asked for the far end. If your answer lies between the two points you started with, you have solved the wrong problem.
Writing 2A − P instead of 2P − A.
The centre is the one that gets doubled. Test the shortcut on an easy case — A(0, 0) about P(1, 1) must give (2, 2).
Multiplying only part of the equation by 2.
(−3 + s)/2 = 2 becomes −3 + s = 4, not −3 + s = 2. The whole of the other side doubles.
Mixing up which point is the centre.
Some questions call the centre A and the point P. Read the sentence, not the letters.
Not checking.
Average your answer with the original — you must land exactly on the centre. Five seconds, and it catches everything.
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