Step by step
The question
Question: rotate the point P(3, 1) 180° about the point (1, 2).
1 Draw the shape
Put the point on the grid.
The center of the turn is (1, 2).
Everything turns about this point.
2 Turn one point
From the center to P: go right 2
then down 1.
Now turn 180° about the center.
The path from the center turns too.
Right 2 turns into left 2.
Down 1 turns into up 1.
We land on P', (−1, 3).
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.
The turn rules we know turn about the origin.
So first we move the center to the origin: take (1, 2) away from P.
P is (2, −1) from the center.
Turn 180°: the rule (x, y) goes to (−x, −y).
(2, −1) becomes (−2, 1).
Then move back: add (1, 2).
We get (−1, 3): P', the same as on the grid!
Careful!
That is a turn about the origin.
The center here is (1, 2), so we take it away first and add it back at the end.
So the image is P'(−1, 3).
4 Check
Let's check P.
Join the center to P
and the center to P'.
From the center, P is (2, −1) away and P' is (−2, 1) away.
Both have length √5: the same distance, as a turn must keep.
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.