Enlargements

What Is an Enlargement?

An enlargement is a transformation that changes the size of a shape.

• The shape stays the same (angles and proportions are unchanged)
• Only the size and position of the shape change
• The shape is scaled from a fixed point called the centre of enlargement

Key Facts:

• If the scale factor is greater than 1, the shape gets bigger
• If the scale factor is between 0 and 1, the shape shrinks
• If the scale factor is negative, the shape is enlarged and rotated $180\degree$ about the centre of enlargement

How To Enlarge a Shape

Step-by-Step

Step 1: Mark the centre of enlargement on the grid.

Step 2: For each vertex on the original shape, measure the horizontal and vertical distance from the centre.

Step 3: Multiply both distances by the scale factor.

Step 4: Plot the new points by measuring the scaled distances from the centre.

Step 5: Join the new vertices and label the enlarged shape.

Example

Question:

Enlarge the triangle Q using a scale factor of 2 and centre of enlargement (0, 0).

Solution:

We multiply the position of each point from the origin by 2:

• $A(1, 1) \rightarrow A'(2, 2)$
• $B(3, 1) \rightarrow B'(6, 2)$
• $C(2, 3) \rightarrow C'(4, 6)$

Plot the new triangle with vertices $A', B', C'$ and label it clearly.

Example

Question:

Shape Q is enlarged by scale factor $\frac{1}{2}$ from centre $(2, 3)$.

Find the enlarged shape.

 
Solution:

Describing an Enlargement

To describe an enlargement, include:

1. Type of transformation: Enlargement
2. Scale factor: The multiplier for each length
3. Centre of enlargement: Coordinates of the fixed point

Example:

"An enlargement by scale factor 3 with centre (0, 0)"

Negative Enlargements

Negative scale factors:

• Still change the size of the shape
• Also rotate it by $180\degree$ around the centre
• Measure distances through the centre in the opposite direction

Example

Question:

Enlarge shape R with vertices $(3, 2), (4, 2), (4, 4), (3, 4)$ by scale factor $−1$ about $(0, 0)$

Solution:

Multiply each coordinate by $-1$:

• $(3, 2) \rightarrow (-3, -2)$
• $(4, 2) \rightarrow (-4, -2)$
• $(4, 4) \rightarrow (-4, -4)$
• $(3, 4) \rightarrow (-3, -4)$

The shape is the same size, rotated $180\degree$.

Area in Enlargements

To find the area scale factor, square the length scale factor.

If:

Scale factor = $k$

Area of original = $A$

Area of enlarged = $A \times k^2$

Example:

If the scale factor is $\frac{1}{2}$ and the original area is $36\text{ cm}^2$, then: $\text{New area} = 36 \times \left(\frac{1}{2}\right)^2 = 36 \times \frac{1}{4} = 9\text{ cm}^2$