Step by step
The question
Question: rotate the shape A(3, 1), B(5, 1) and C(3, 4) 90° counterclockwise about the point (2, 1).
1 Draw the shape
Put the points on the grid and join them.
The center of the turn is (2, 1).
Everything turns about this point.
2 Turn one point
Counterclockwise means the other way from the hands of a clock.
From the center to C: go right 1
then up 3.
Now turn 90° counterclockwise about the center.
The path from the center turns too.
Right 1 turns into up 1.
Up 3 turns into left 3.
We land on C', (−1, 2).
We call the new point C prime.
The little mark means: the new C.
3 Find the rule
Now the same with numbers, for C.
The turn rules we know turn about the origin.
So first we move the center to the origin: take (2, 1) away from C.
C is (1, 3) from the center.
Turn 90° counterclockwise: the rule (x, y) goes to (−y, x).
(1, 3) becomes (−3, 1).
Then move back: add (2, 1).
We get (−1, 2): C', the same as on the grid!
Careful!
That is a turn about the origin.
The center here is (2, 1), so we take it away first and add it back at the end.
4 Turn the other points
A: take away the center, (1, 0).
Turn: (0, 1).
Add the center back: (2, 2).
B: take away the center, (3, 0).
Turn: (0, 3).
Add the center back: (2, 4).
Join the new points.
This new shape is called the image.
So the image is A'(2, 2), B'(2, 4) and C'(−1, 2).
5 Check
Let's check C.
Join the center to C
and the center to C'.
From the center, C is (1, 3) away and C' is (−3, 1) away.
Both have length √10: the same distance, as a turn must keep.
They make a right angle: a quarter turn.
A quarter turn, counterclockwise.
Right!
Let's look back.
First: draw the shape.
Then: turn one point.
Then: find the rule.
Then: turn the other points.
Last: check.