Step by step
The question
Question: rotate the point P(2, 4) 90° clockwise about the point (1, 1).
1 Draw the shape
Put the point on the grid.
The center of the turn is (1, 1).
Everything turns about this point.
2 Turn one point
Clockwise means the way the hands of a clock turn.
From the center to P: go right 1
then up 3.
Now turn 90° clockwise about the center.
The path from the center turns too.
Right 1 turns into down 1.
Up 3 turns into right 3.
We land on P', (4, 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.
The turn rules we know turn about the origin.
So first we move the center to the origin: take (1, 1) away from P.
P is (1, 3) from the center.
Turn 90° clockwise: the rule (x, y) goes to (y, −x).
(1, 3) becomes (3, −1).
Then move back: add (1, 1).
We get (4, 0): P', the same as on the grid!
Careful!
That is a turn about the origin.
The center here is (1, 1), so we take it away first and add it back at the end.
So the image is P'(4, 0).
4 Check
Let's check P.
Join the center to P
and the center to P'.
From the center, P is (1, 3) away and P' is (3, −1) away.
Both have length √10: the same distance, as a turn must keep.
They make a right angle: a quarter turn.
A quarter turn, clockwise.
Right!
Let's look back.
First: draw the shape.
Then: turn one point.
Then: find the rule.
Last: check.