Step by step
The question
Question: rotate the shape A(0, 0), B(2, 0) and C(0, 2) 180° 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
From the center to A: go right 1
then down 1.
Now turn 180° about the center.
The path from the center turns too.
Right 1 turns into left 1.
Down 1 turns into up 1.
We land on A', (−2, 2).
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 180°: the rule (x, y) goes to (−x, −y).
(1, −1) becomes (−1, 1).
Then move back: add (−1, 1).
We get (−2, 2): 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: (−3, 1).
Add the center back: (−4, 2).
C: take away the center, (1, 1).
Turn: (−1, −1).
Add the center back: (−2, 0).
Join the new points.
This new shape is called the image.
So the image is A'(−2, 2), B'(−4, 2) and C'(−2, 0).
5 Check
Let's check B.
Join the center to B
and the center to B'.
From the center, B is (3, −1) away and B' is (−3, 1) away.
Both have length √10: the same distance, as a turn must keep.
And B, the center and B' are on one straight line: a half turn.
Right!
Let's look back.
First: draw the shape.
Then: turn one point.
Then: find the rule.
Then: turn the other points.
Last: check.