Step by step
The question
Question: rotate the point P(5, 2) 180° about the point (2, 1).
1 Draw the shape
Put the point on the grid.
The center of the turn is (2, 1).
Everything turns about this point.
2 Turn one point
From the center to P: go right 3
then up 1.
Now turn 180° about the center.
The path from the center turns too.
Right 3 turns into left 3.
Up 1 turns into down 1.
We land on P': from the center (2, 1), left 3, down 1.
That is (−1, 0).
We call the new point P prime.
The little mark means: the new P.
3 Find the rule
Now the same with numbers, for P.
Turn rules work about the origin (0, 0).
So first we move the center to the origin: take (2, 1) away from P.
We get (3, 1): the same as right 3, up 1 on the grid.
Turn 180°.
On the grid, right 3, up 1 became left 3, down 1.
So (3, 1) becomes (−3, −1).
That is the turn rule: (x, y) goes to (−x, −y).
Then move back: add (2, 1).
We get (−1, 0): P', the same as on the grid!
Careful!
That is a turn about the origin.
The center here is (2, 1), so we take it away first and add it back at the end.
So the image is P'(−1, 0).
4 Check
Let's check P.
Join the center to P
and the center to P'.
From the center to P: right 3, up 1.
To P': left 3, down 1.
The same step sizes, 1 and 3, just turned.
So P' is just as far from the center as P.
A turn keeps the distance.
And P, the center and P' are on one straight line: a half turn.
Right!
Let's look back.
First: draw the shape.
Then: turn one point.
Then: find the rule.
Last: check.