Step by step
The question
Question: rotate the shape A(1, −1), B(3, −1) and C(1, 1) 270° counterclockwise about the point (−1, −2).
1 Draw the shape
Put the points on the grid and join them.
The center of the turn is (−1, −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 A: go right 2
then up 1.
Now turn 270° counterclockwise about the center.
The path from the center turns too.
Right 2 turns into down 2.
Up 1 turns into right 1.
We land on A', (0, −4).
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, −2) away from A.
A is (2, 1) from the center.
Turn 270° counterclockwise: the rule (x, y) goes to (y, −x).
(2, 1) becomes (1, −2).
Then move back: add (−1, −2).
We get (0, −4): A', 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.
4 Turn the other points
B: take away the center, (4, 1).
Turn: (1, −4).
Add the center back: (0, −6).
C: take away the center, (2, 3).
Turn: (3, −2).
Add the center back: (2, −4).
Join the new points.
This new shape is called the image.
So the image is A'(0, −4), B'(0, −6) and C'(2, −4).
5 Check
Let's check B.
Join the center to B
and the center to B'.
From the center, B is (4, 1) away and B' is (1, −4) away.
Both have length √17: 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.
Then: turn the other points.
Last: check.