Step by step
The question
Question: rotate the shape A(1, 1), B(3, 1) and C(3, 2) 180° about the point (2, 3).
1 Draw the shape
Put the points on the grid and join them.
The center of the turn is (2, 3).
Everything turns about this point.
2 Turn one point
From the center to A: go left 1
then down 2.
Now turn 180° about the center.
The path from the center turns too.
Left 1 turns into right 1.
Down 2 turns into up 2.
We land on A', (3, 5).
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, 3) away from A.
A is (−1, −2) from the center.
Turn 180°: the rule (x, y) goes to (−x, −y).
(−1, −2) becomes (1, 2).
Then move back: add (2, 3).
We get (3, 5): A', the same as on the grid!
Careful!
That is a turn about the origin.
The center here is (2, 3), 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: (−1, 2).
Add the center back: (1, 5).
C: take away the center, (1, −1).
Turn: (−1, 1).
Add the center back: (1, 4).
Join the new points.
This new shape is called the image.
So the image is A'(3, 5), B'(1, 5) and C'(1, 4).
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 (1, 2) away.
Both have length √5: the same distance, as a turn must keep.
And A, the center and A' 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.