Step by step
The question
Question: rotate the shape A(2, 0), B(4, 0) and C(2, 3) 90° counterclockwise about the point (1, −1).
1 Draw the shape
Put the points on the grid and join them.
The center of the turn is (1, −1).
Everything turns about this point.
2 Turn one point
Counterclockwise means the other way from the hands of a clock.
From the center to A: go right 1
then up 1.
Now turn 90° counterclockwise about the center.
The path from the center turns too.
Right 1 turns into up 1.
Up 1 turns into left 1.
We land on A', (0, 0).
We call the new point A prime.
The little mark means: the new A.
3 Find the rule
Now the same with numbers, for A.
The turn rules we know turn about the origin.
So first we move the center to the origin: take (1, −1) away from A.
A is (1, 1) from the center.
Turn 90° counterclockwise: the rule (x, y) goes to (−y, x).
(1, 1) becomes (−1, 1).
Then move back: add (1, −1).
We get (0, 0): A', 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.
4 Turn the other points
B: take away the center, (3, 1).
Turn: (−1, 3).
Add the center back: (0, 2).
C: take away the center, (1, 4).
Turn: (−4, 1).
Add the center back: (−3, 0).
Join the new points.
This new shape is called the image.
So the image is A'(0, 0), B'(0, 2) and C'(−3, 0).
5 Check
Let's check C.
Join the center to C
and the center to C'.
From the center, C is (1, 4) away and C' is (−4, 1) away.
Both have length √17: 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.
Then: turn the other points.
Last: check.