Rotate (3, 1) 180° about the origin Easy

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.

▶ Watch it solved

Rotate A(3, 1) B(4, 3) C(1, 2) 90° counterclockwise about the origin Medium

The question

Question: rotate the shape A(3, 1), B(4, 3) and C(1, 2) 90° counterclockwise about the origin.

1 Draw the shape

Put the points on the grid and join them.

2 Turn one point

Counterclockwise means the other way from the hands of a clock.

Start at the center O.

To reach A: go right 3

then up 1.

Now turn a quarter turn counterclockwise about O.

Everything turns with it.

Right 3 turns into up 3.

Up 1 turns into left 1.

We land on the new point A'.

We call the new point A prime.

The little mark means: the new A.

3 Find the rule

Look at the numbers of A and A'.

A(3, 1) goes to A'(−1, 3).

Careful!

A quarter turn swaps the two numbers.

Changing a sign alone is a flip, not a turn.

The two numbers swap places, and the new first number gets a minus sign.

So the rule is: (x, y) goes to (−y, x).

4 Turn the other points

B is (4, 3): swap and change the sign of the first: B' is (−3, 4).

C is (1, 2): swap and change the sign of the first: C' is (−2, 1).

Join the new points.

This new shape is called the image.

So the image is A'(−1, 3), B'(−3, 4) and C'(−2, 1).

5 Check

Let's check B on the grid.

Join O to B

and O to B'.

They make a right angle: a quarter turn.

And it turns 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.

▶ Watch it solved

Rotate A(2, 1) B(4, 1) C(4, 3) 270° counterclockwise about the origin Harder

The question

Question: rotate the shape A(2, 1), B(4, 1) and C(4, 3) 270° counterclockwise about the origin.

1 Draw the shape

Put the points on the grid and join them.

2 Turn one point

Counterclockwise means the other way from the hands of a clock.

Three quarter turns counterclockwise end in the same place as one quarter turn clockwise.

Start at the center O.

To reach A: go right 2

then up 1.

Now turn three quarter turns counterclockwise about O.

Everything turns with it.

Right 2 turns into down 2.

Up 1 turns into right 1.

We land on the new point A'.

We call the new point A prime.

The little mark means: the new A.

3 Find the rule

Look at the numbers of A and A'.

A(2, 1) goes to A'(1, −2).

Careful!

A quarter turn swaps the two numbers.

Changing a sign alone is a flip, not a turn.

The two numbers swap places, and the new second number gets a minus sign.

So the rule is: (x, y) goes to (y, −x).

4 Turn the other points

B is (4, 1): swap and change the sign of the second: B' is (1, −4).

C is (4, 3): swap and change the sign of the second: C' is (3, −4).

Join the new points.

This new shape is called the image.

So the image is A'(1, −2), B'(1, −4) and C'(3, −4).

5 Check

Let's check B on the grid.

Join O to B

and O to B'.

They make a right angle: a quarter turn.

And it turns clockwise.

Right!

Let's look back.

First: draw the shape.

Then: turn one point.

Then: find the rule.

Then: turn the other points.

Last: check.

▶ Watch it solved