Step by step
The question
Question: rotate the shape A(1, 2), B(4, 2) and C(4, 4) 180° about the point (1, 0).
1 Draw the shape
Put the points on the grid and join them.
The center of the turn is (1, 0).
Everything turns about this point.
2 Turn one point
From the center to B: go right 3
then up 2.
Now turn 180° about the center.
The path from the center turns too.
Right 3 turns into left 3.
Up 2 turns into down 2.
We land on B', (−2, −2).
We call the new point B prime.
The little mark means: the new B.
3 Find the rule
Now the same with numbers, for B.
The turn rules we know turn about the origin.
So first we move the center to the origin: take (1, 0) away from B.
B is (3, 2) from the center.
Turn 180°: the rule (x, y) goes to (−x, −y).
(3, 2) becomes (−3, −2).
Then move back: add (1, 0).
We get (−2, −2): B', the same as on the grid!
Careful!
That is a turn about the origin.
The center here is (1, 0), so we take it away first and add it back at the end.
4 Turn the other points
A: take away the center, (0, 2).
Turn: (0, −2).
Add the center back: (1, −2).
C: take away the center, (3, 4).
Turn: (−3, −4).
Add the center back: (−2, −4).
Join the new points.
This new shape is called the image.
So the image is A'(1, −2), B'(−2, −2) and C'(−2, −4).
5 Check
Let's check C.
Join the center to C
and the center to C'.
From the center, C is (3, 4) away and C' is (−3, −4) away.
Both have length √25: the same distance, as a turn must keep.
And C, the center and C' 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.