Step by step
The question
Question: rotate the shape A(−3, −1), B(−1, −1) and C(−1, 1) 90° counterclockwise about the point (−2, 1).
1 Draw the shape
Put the points on the grid and join them.
The center of the turn is (−2, 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 left 1
then down 2.
Now turn 90° counterclockwise about the center.
The path from the center turns too.
Left 1 turns into down 1.
Down 2 turns into right 2.
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 (−2, 1) away from A.
A is (−1, −2) from the center.
Turn 90° counterclockwise: the rule (x, y) goes to (−y, x).
(−1, −2) becomes (2, −1).
Then move back: add (−2, 1).
We get (0, 0): A', 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.
4 Turn the other points
B: take away the center, (1, −2).
Turn: (2, 1).
Add the center back: (0, 2).
C: take away the center, (1, 0).
Turn: (0, 1).
Add the center back: (−2, 2).
Join the new points.
This new shape is called the image.
So the image is A'(0, 0), B'(0, 2) and C'(−2, 2).
5 Check
Let's check A.
Join the center to A
and the center to A'.
From the center, A is (−1, −2) away and A' is (2, −1) away.
Both have length √5: 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.