Introduction to Enlargement and Scale Factor

TL;DR

Enlargement makes a shape bigger or smaller, but keeps its proportions the same. The scale factor tells you how many times bigger or smaller the new shape is. You find the new size by multiplying the old size by the scale factor.

1. The Mental Model

Imagine you have a photocopying machine that can make copies bigger or smaller. Enlargement is like using that machine. The original shape is your document, and the new shape is the copy.

2. The Core Material

When you enlarge a shape, you create a new shape that's mathematically similar to the original. This means that:
* All corresponding angles stay exactly the same.
* All corresponding side lengths are multiplied by the same amount.

What is a Scale Factor?

The scale factor is simply the number that tells you how much bigger or smaller the enlarged shape is compared to the original.

• If the scale factor is greater than 1 (e.g., 2, 3.5, 10), the new shape will be bigger than the original.
• If the scale factor is between 0 and 1 (e.g., 0.5, 1/2, 0.25), the new shape will be smaller than the original (this is often called a reduction).
• If the scale factor is 1, the new shape is exactly the same size as the original (no enlargement).

How to use the Scale Factor

To find the new length of any side in the enlarged shape, you just multiply the original side length by the scale factor.

New Length = Original Length × Scale Factor

Centre of Enlargement (Briefly)

While we won't draw enlargements yet, it's good to know that every enlargement has a centre of enlargement. This is a fixed point from which all points on the original shape are enlarged. We'll dive into this more later, but for now, just know it exists.

3. Worked Example

Let's say you have a rectangle, ABCD, with a width of 4 cm and a height of 2 cm. You want to enlarge this rectangle by a scale factor of 3.

1. Identify original dimensions:

   • Original width = 4 cm
   • Original height = 2 cm

2. Identify the scale factor:

   • Scale factor = 3

3. Calculate new dimensions:

   • New width = Original width × Scale factor = 4 cm × 3 = 12 cm
   • New height = Original height × Scale factor = 2 cm × 3 = 6 cm

So, the enlarged rectangle will have a width of 12 cm and a height of 6 cm. All its angles will still be 90 degrees.

4. Key Takeaways

• Enlargement creates a new shape that is similar to the original.
• All angles in the enlarged shape remain the same as the original.
• The scale factor tells you how much bigger or smaller the new shape is.
• Multiply original side lengths by the scale factor to find new side lengths.
• A scale factor greater than 1 means the shape gets bigger.
• A scale factor between 0 and 1 means the shape gets smaller (a reduction).
• There's always a 'centre of enlargement' even when you're just calculating sizes.

Common mistakes to avoid:
- Don't add or subtract the scale factor; always multiply.
- Don't change the angles of the shape; they always stay the same.
- Forgetting that a scale factor less than 1 makes the shape smaller.
- Only applying the scale factor to one side; apply it to all sides.

5. Now Try It

Imagine you have a triangle with sides measuring 5 cm, 7 cm, and 8 cm. Enlarge this triangle using a scale factor of 2.5. What will be the lengths of the three sides of the new, enlarged triangle? List all three new side lengths.