Worked examples
The question
Question: dilate the point P(2, 3) by scale factor 2.
P = (2, 3)
= scale factor 2
1 Draw the shape
Put the point on the grid.
2 Multiply each point
Scale factor 2 makes the shape 2 times as big.
Every number is multiplied by 2.
scale factor 2 → 2 times as big
We call the new point P prime.
The little mark means: the new P.
P is (2, 3).
2 times 2 is 4.
2 times 3 is 6.
So P' is (4, 6).
P(2, 3) → (2 × 2, 2 × 3) = P'(4, 6)
Careful!
A dilation multiplies.
Adding 2 would slide the shape, not make it bigger.
Careful: not P'(2 + 2, 3 + 2) ✗
3 The new point
So the image is P'(4, 6).
P' = (4, 6)
4 Check
Let's check P.
Draw a line from O through P to P'.
From O to P: right 2, up 3.
O → P: right 2, up 3
From O to P': right 4, up 6.
That is 2 times the moves to P.
So P' is on the same line from O, 2 times as far.
Right!
O → P': right 4, up 6
Let's look back.
First: draw the shape.
Then: multiply each point.
Then: the new point.
Last: check.
▶ Watch it solved
The question
Question: dilate the shape A(2, 1), B(3, 1) and C(2, 3) by scale factor 3.
A(2, 1), B(3, 1) and C(2, 3)
= scale factor 3
1 Draw the shape
Put the points on the grid and join them.
2 Multiply each point
Scale factor 3 makes the shape 3 times as big.
Every number is multiplied by 3.
scale factor 3 → 3 times as big
We call the new point A prime.
The little mark means: the new A.
A is (2, 1).
3 times 2 is 6.
3 times 1 is 3.
So A' is (6, 3).
A(2, 1) → (3 × 2, 3 × 1) = A'(6, 3)
Careful!
A dilation multiplies.
Adding 3 would slide the shape, not make it bigger.
Careful: not A'(2 + 3, 1 + 3) ✗
B: 3 times 3 is 9, 3 times 1 is 3.
So B' is (9, 3).
B(3, 1) → (3 × 3, 3 × 1) = B'(9, 3)
C: 3 times 2 is 6, 3 times 3 is 9.
So C' is (6, 9).
C(2, 3) → (3 × 2, 3 × 3) = C'(6, 9)
3 Draw the image
Join the new points.
This new shape is called the image.
So the image is A'(6, 3), B'(9, 3) and C'(6, 9).
A'(6, 3), B'(9, 3) and C'(6, 9)
4 What changes
Count the flat side AB on the grid: 1 squares.
AB = 1
A'B' is 3 squares.
That is 3 times as long.
Every length is multiplied by 3.
A'B' = 3 = 3 × 1
The angles do not change.
The image is the same shape, just bigger.
angles stay the same
5 Check
Let's check A.
Draw a line from O through A to A'.
From O to A: right 2, up 1.
O → A: right 2, up 1
From O to A': right 6, up 3.
That is 3 times the moves to A.
So A' is on the same line from O, 3 times as far.
Right!
O → A': right 6, up 3
Let's look back.
First: draw the shape.
Then: multiply each point.
Then: draw the image.
Then: what changes.
Last: check.
Answer A'(6, 3), B'(9, 3) and C'(6, 9)
▶ Watch it solved
The question
Question: dilate the shape A(1, 1), B(3, 1), C(3, 2) and D(1, 2) by scale factor 2.
A(1, 1), B(3, 1), C(3, 2) and D(1, 2)
= scale factor 2
1 Draw the shape
Put the points on the grid and join them.
2 Multiply each point
Scale factor 2 makes the shape 2 times as big.
Every number is multiplied by 2.
scale factor 2 → 2 times as big
We call the new point A prime.
The little mark means: the new A.
A is (1, 1).
2 times 1 is 2.
2 times 1 is 2.
So A' is (2, 2).
A(1, 1) → (2 × 1, 2 × 1) = A'(2, 2)
Careful!
A dilation multiplies.
Adding 2 would slide the shape, not make it bigger.
Careful: not A'(1 + 2, 1 + 2) ✗
B: 2 times 3 is 6, 2 times 1 is 2.
So B' is (6, 2).
B(3, 1) → (2 × 3, 2 × 1) = B'(6, 2)
C: 2 times 3 is 6, 2 times 2 is 4.
So C' is (6, 4).
C(3, 2) → (2 × 3, 2 × 2) = C'(6, 4)
D: 2 times 1 is 2, 2 times 2 is 4.
So D' is (2, 4).
D(1, 2) → (2 × 1, 2 × 2) = D'(2, 4)
3 Draw the image
Join the new points.
This new shape is called the image.
So the image is A'(2, 2), B'(6, 2), C'(6, 4) and D'(2, 4).
A'(2, 2), B'(6, 2), C'(6, 4) and D'(2, 4)
4 What changes
Count the flat side AB on the grid: 2 squares.
AB = 2
A'B' is 4 squares.
That is 2 times as long.
Every length is multiplied by 2.
A'B' = 4 = 2 × 2
The angles do not change.
The image is the same shape, just bigger.
angles stay the same
5 Check
Let's check A.
Draw a line from O through A to A'.
From O to A: right 1, up 1.
O → A: right 1, up 1
From O to A': right 2, up 2.
That is 2 times the moves to A.
So A' is on the same line from O, 2 times as far.
Right!
O → A': right 2, up 2
Let's look back.
First: draw the shape.
Then: multiply each point.
Then: draw the image.
Then: what changes.
Last: check.
Answer A'(2, 2), B'(6, 2), C'(6, 4) and D'(2, 4)
▶ Watch it solved
