Step by step
The question
Question: rotate the point P(3, 1) 180° about the origin.
1 Draw the shape
Put the point on the grid.
2 Turn one point
Start at the center O.
To reach P: 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 P'.
We call the new point P prime.
The little mark means: the new P.
3 Find the rule
Look at the numbers of P and P'.
P(3, 1) goes to P'(−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).
So the image is P'(−3, −1).
4 Check
Let's check P on the grid.
Join O to P
and O to P'.
P, O and P' 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.
Last: check.