Step by step
The question
Question: rotate the shape A(1, 1), B(3, 1) and C(1, 4) 180° about the origin.
1 Draw the shape
Put the points on the grid and join them.
2 Turn one point
Start at the center O.
To reach B: go right 3
then up 1.
Now turn a half turn about O.
Everything turns with it.
Right 3 turns into left 3.
Up 1 turns into down 1.
We land on the new point B'.
We call the new point B prime.
The little mark means: the new B.
3 Find the rule
Look at the numbers of B and B'.
B(3, 1) goes to B'(−3, −1).
Careful!
A half turn changes both signs.
Changing one sign is a flip, not a turn.
Both numbers stay in their places, and both change sign.
So the rule is: (x, y) goes to (−x, −y).
4 Turn the other points
A is (1, 1): both signs change, so A' is (−1, −1).
C is (1, 4): both signs change, so C' is (−1, −4).
Join the new points.
This new shape is called the image.
So the image is A'(−1, −1), B'(−3, −1) and C'(−1, −4).
5 Check
Let's check A on the grid.
Join O to A
and O to A'.
A, O and A' are on one straight line, with O in the middle.
That is 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.