Understanding Scaled Copies and Scale Factors

TL;DR

A scaled copy is a new figure that looks exactly like the original but is bigger or smaller. Every length in a scaled copy is multiplied by the same number, called the scale factor. If the scale factor is 1, the copy is the same size as the original.

1. The Mental Model

Imagine taking a photo on your phone and then zooming in or out. The zoomed-in or zoomed-out picture is a scaled copy of the original. Everything in the picture stretches or shrinks uniformly.

2. The Core Material

When you make a scaled copy of a shape, you're creating a new shape that has the exact same angles as the original, but its side lengths are all proportionally changed. Think of it like using a photocopier to enlarge or reduce an image.

What Makes a Scaled Copy?

Photo by https://kaboompics.com/ on Pexels

For a figure to be a scaled copy of another, two things must be true:
1. All corresponding angles must be equal. If the original has a 90-degree corner, the scaled copy must also have a 90-degree corner in the same spot.
2. All corresponding side lengths must be multiplied by the same number. This "same number" is what we call the scale factor.

The Scale Factor

Photo by Caroline Cagnin on Pexels

The scale factor tells you how much bigger or smaller the new figure is compared to the original.

• If the scale factor is greater than 1, the scaled copy is larger than the original.
• If the scale factor is between 0 and 1 (a fraction or decimal), the scaled copy is smaller than the original.
• If the scale factor is exactly 1, the scaled copy is the same size as the original.

To find the scale factor, you always divide a length from the new figure by the corresponding length from the original figure:

Scale Factor = (Length on Scaled Copy) / (Corresponding Length on Original)

It's super important to keep the order right! New over original.

How to Create a Scaled Copy

Photo by https://kaboompics.com/ on Pexels

graph TD
    A["Choose Original Figure"] --> B["Identify all side lengths and angles"]
    B --> C{"Choose a Scale Factor (k)"}
    C --> D["Multiply EACH original side length by 'k'"]
    D --> E["Keep ALL original angles the same"]
    E --> F["Draw the new figure using modified side lengths and original angles"]
    F --> G["Verify: All angles are same, all sides are proportional"]

Example: Finding Missing Side Lengths

Photo by Bernice Chan on Pexels

Let's say you have a rectangle with sides 4 cm and 6 cm. You make a scaled copy with a scale factor of 2.5.
* New width: 4 cm * 2.5 = 10 cm
* New length: 6 cm * 2.5 = 15 cm
The angles (all 90 degrees) stay the same.

3. Worked Example

You have a triangle ABC with sides AB = 3 units, BC = 4 units, and AC = 5 units. Its angles are 37°, 53°, and 90°.
You are given a scaled copy, triangle DEF, where DE corresponds to AB, EF to BC, and DF to AC. You know that DE = 6 units.

Problem: Find the scale factor and the lengths of EF and DF.

Step 1: Find the scale factor.
We know DE corresponds to AB.
Scale Factor = (Length on Scaled Copy) / (Corresponding Length on Original)
Scale Factor = DE / AB = 6 units / 3 units = 2.

Step 2: Find the length of EF.
EF corresponds to BC.
EF = BC * Scale Factor
EF = 4 units * 2 = 8 units.

Step 3: Find the length of DF.
DF corresponds to AC.
DF = AC * Scale Factor
DF = 5 units * 2 = 10 units.

So, the scale factor is 2, and the sides of the scaled copy are 6, 8, and 10 units. The angles in triangle DEF would still be 37°, 53°, and 90°.

4. Key Takeaways

• A scaled copy has the same shape but a different size (unless the scale factor is 1).
• All angles in a scaled copy are identical to the original figure's corresponding angles.
• All side lengths in a scaled copy are multiplied by the same number, called the scale factor.
• To find the scale factor, divide a length from the scaled copy by its corresponding length from the original.
• A scale factor greater than 1 makes the copy bigger; less than 1 makes it smaller.
• The order matters when calculating scale factor: new length / original length.

Common Mistakes to Avoid:
- Mixing up original and copy: Always remember new / original for scale factor.
- Changing angles: Only side lengths change; angles stay the same in a scaled copy.
- Applying different factors: You must multiply all side lengths by the same scale factor.
- Assuming it's a scaled copy: Check both angles and proportional side lengths before concluding it's a scaled copy.

5. Now Try It

Draw a simple triangle with side lengths 3 cm, 4 cm, and 5 cm (a right-angled triangle is easiest). Then, create a scaled copy of this triangle using a scale factor of 1.5. Label the sides of your original and your new scaled copy. What are the new side lengths?
Success looks like a new triangle where each side length is 1.5 times its original, and its angles are identical to the original triangle's angles.