Step by step
The question
Question: rotate the point P(5, 2) 90° counterclockwise about the point (2, 2).
1 Draw the shape
Put the point on the grid.
The center of the turn is (2, 2).
Everything turns about this point.
2 Turn one point
Counterclockwise means the other way from the hands of a clock.
From the center to P: go right 3
Now turn 90° counterclockwise about the center.
The path from the center turns too.
Right 3 turns into up 3.
We land on P', (2, 5).
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 (2, 2) away from P.
P is (3, 0) from the center.
Turn 90° counterclockwise: the rule (x, y) goes to (−y, x).
(3, 0) becomes (0, 3).
Then move back: add (2, 2).
We get (2, 5): P', the same as on the grid!
Careful!
That is a turn about the origin.
The center here is (2, 2), so we take it away first and add it back at the end.
So the image is P'(2, 5).
4 Check
Let's check P.
Join the center to P
and the center to P'.
From the center, P is (3, 0) away and P' is (0, 3) away.
Both have length √9: the same distance, as a turn must keep.
They make a right angle: a quarter turn.
A quarter turn, counterclockwise.
Right!
Let's look back.
First: draw the shape.
Then: turn one point.
Then: find the rule.
Last: check.