Worked examples
The question
Question: rotate the point P(4, 3) 180° about the point (1, 1).
P = (4, 3)
= 180° about (1, 1)
1 Draw the shape
Put the point on the grid.
The center of the turn is (1, 1).
Everything turns about this point.
2 Turn one point
From the center to P: 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 P', (−2, −1).
We call the new point P prime.
The little mark means: the new P.
3 Find the rule
Now the same with numbers, for P.
The turn rules we know turn about the origin.
So first we move the center to the origin: take (1, 1) away from P.
P is (3, 2) from the center.
1. take away the center: (4 − 1, 3 − 1) = (3, 2)
Turn 180°: the rule (x, y) goes to (−x, −y).
(3, 2) becomes (−3, −2).
2. turn: (x, y) → (−x, −y): (3, 2) → (−3, −2)
Then move back: add (1, 1).
We get (−2, −1): P', the same as on the grid!
3. add the center back: (−3 + 1, −2 + 1) = (−2, −1)
Careful!
That is a turn about the origin.
The center here is (1, 1), so we take it away first and add it back at the end.
Careful: not P'(−4, −3) ✗
So the image is P'(−2, −1).
P' = (−2, −1)
4 Check
Let's check P.
Join the center to P
and the center to P'.
From the center, P is (3, 2) away and P' is (−3, −2) away.
Both have length √13: the same distance, as a turn must keep.
center → P: (3, 2), center → P': (−3, −2): both √13 long
And P, the center and P' are on one straight line: a half turn.
Right!
P, center, P': one straight line
Let's look back.
First: draw the shape.
Then: turn one point.
Then: find the rule.
Last: check.
▶ Watch it solved
The question
Question: rotate the shape A(3, 1), B(5, 1) and C(3, 4) 90° counterclockwise about the point (2, 1).
A(3, 1), B(5, 1) and C(3, 4)
= 90° counterclockwise about (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.
counterclockwise = against the 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.
1. take away the center: (3 − 2, 4 − 1) = (1, 3)
Turn 90° counterclockwise: the rule (x, y) goes to (−y, x).
(1, 3) becomes (−3, 1).
2. turn: (x, y) → (−y, x): (1, 3) → (−3, 1)
Then move back: add (2, 1).
We get (−1, 2): C', the same as on the grid!
3. add the center back: (−3 + 2, 1 + 1) = (−1, 2)
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.
Careful: not C'(−4, 3) ✗
4 Turn the other points
A: take away the center, (1, 0).
Turn: (0, 1).
Add the center back: (2, 2).
A: (1, 0) → (0, 1) → A'(2, 2)
B: take away the center, (3, 0).
Turn: (0, 3).
Add the center back: (2, 4).
B: (3, 0) → (0, 3) → B'(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).
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.
center → C: (1, 3), center → C': (−3, 1): both √10 long
They make a right angle: a quarter turn.
A quarter turn, counterclockwise.
Right!
a right angle at the center: a quarter turn
Let's look back.
First: draw the shape.
Then: turn one point.
Then: find the rule.
Then: turn the other points.
Last: check.
Answer A'(2, 2), B'(2, 4) and C'(−1, 2)
▶ Watch it solved
The question
Question: rotate the shape A(1, −1), B(3, −1) and C(1, 1) 270° counterclockwise about the point (−1, −2).
A(1, −1), B(3, −1) and C(1, 1)
= 270° counterclockwise about (−1, −2)
1 Draw the shape
Put the points on the grid and join them.
The center of the turn is (−1, −2).
Everything turns about this point.
2 Turn one point
Counterclockwise means the other way from the hands of a clock.
counterclockwise = against the clock
From the center to A: go right 2
then up 1.
Now turn 270° counterclockwise about the center.
The path from the center turns too.
Right 2 turns into down 2.
Up 1 turns into right 1.
We land on A', (0, −4).
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 (−1, −2) away from A.
A is (2, 1) from the center.
1. take away the center: (1 − (−1), −1 − (−2)) = (2, 1)
Turn 270° counterclockwise: the rule (x, y) goes to (y, −x).
(2, 1) becomes (1, −2).
2. turn: (x, y) → (y, −x): (2, 1) → (1, −2)
Then move back: add (−1, −2).
We get (0, −4): A', the same as on the grid!
3. add the center back: (1 + (−1), −2 + (−2)) = (0, −4)
Careful!
That is a turn about the origin.
The center here is (−1, −2), so we take it away first and add it back at the end.
Careful: not A'(−1, −1) ✗
4 Turn the other points
B: take away the center, (4, 1).
Turn: (1, −4).
Add the center back: (0, −6).
B: (4, 1) → (1, −4) → B'(0, −6)
C: take away the center, (2, 3).
Turn: (3, −2).
Add the center back: (2, −4).
C: (2, 3) → (3, −2) → C'(2, −4)
Join the new points.
This new shape is called the image.
So the image is A'(0, −4), B'(0, −6) and C'(2, −4).
A'(0, −4), B'(0, −6) and C'(2, −4)
5 Check
Let's check B.
Join the center to B
and the center to B'.
From the center, B is (4, 1) away and B' is (1, −4) away.
Both have length √17: the same distance, as a turn must keep.
center → B: (4, 1), center → B': (1, −4): both √17 long
They make a right angle: a quarter turn.
A quarter turn, clockwise.
Right!
a right angle at the center: a quarter turn
Let's look back.
First: draw the shape.
Then: turn one point.
Then: find the rule.
Then: turn the other points.
Last: check.
Answer A'(0, −4), B'(0, −6) and C'(2, −4)
▶ Watch it solved
