How to Find Scale Factor in 3 Easy Steps — Mashup Math - Free Printable

The image you've shared is an educational graphic from Mashup Math that explains how to find the scale factor between two similar figures on a coordinate plane. It provides three examples using triangles and their dilated images, showing how to calculate the scale factor by comparing corresponding coordinates.

Let’s go through each example and explain the solution step-by-step:

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🔷 What is a Scale Factor?

The scale factor is the ratio of the lengths of corresponding sides in two similar figures. When one figure is a dilation (enlargement or reduction) of another, the scale factor tells us how much larger or smaller the new figure is compared to the original.

To find the scale factor:
- Pick a pair of corresponding points from the original and the dilated figure.
- Divide the coordinates of the image point by the coordinates of the original point.
- The result is the scale factor.

> Note: If the scale factor is greater than 1, it's an enlargement; if less than 1, it's a reduction.

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## ✔ Example #1: Enlargement

Original Point B: (2, 3)
Image Point B': (6, 9)

We are given:
> B' x-value ÷ B x-value = 6 ÷ 2 = 3

✔ So, the scale factor is 3.

👉 This means the image triangle is 3 times larger than the original triangle.

💡 Check y-values too: 9 ÷ 3 = 3 → confirms the same scale factor.

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## ✔ Example #2: Reduction

Original Point S: (-8, 8)
Image Point S': (-4, 4)

We are given:
> S' x-value ÷ S x-value = -4 ÷ (-8) = 1/2

✔ So, the scale factor is 1/2.

👉 This means the image triangle is half the size of the original triangle.

💡 Check y-values: 4 ÷ 8 = 1/2 → consistent.

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## ✔ Example #3: Reduction

Original Point R: (9, 3)
Image Point R': (3, 1)

We are given:
> R' x-value ÷ R x-value = 3 ÷ 9 = 1/3

✔ So, the scale factor is 1/3.

👉 The image triangle is one-third the size of the original.

💡 Check y-values: 1 ÷ 3 = 1/3 → consistent.

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📌 Key Takeaways:

1. Scale Factor = Image Coordinate ÷ Original Coordinate (for corresponding points).
2. Use x-values or y-values — both should give the same scale factor if the dilation is centered at the origin.
3. The origin must be the center of dilation for this method to work directly (which appears to be the case here, as all transformations seem centered at (0,0)).
4. If the scale factor is positive, the image is on the same side of the origin; if negative, it's reflected across the origin.

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💡 General Formula:

If point $ P(x, y) $ is dilated to $ P'(x', y') $ with center at the origin:
$$
\text{Scale Factor} = \frac{x'}{x} = \frac{y'}{y}
$$

As long as the dilation is centered at the origin, this method works perfectly.

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✔ Final Answer Summary:

| Example | Scale Factor | Type of Dilation |
|--------|--------------|------------------|
| #1 | 3 | Enlargement |
| #2 | 1/2 | Reduction |
| #3 | 1/3 | Reduction |

Each example correctly demonstrates how to compute the scale factor by dividing the coordinates of the image by the original.

Let me know if you'd like help solving a different problem or applying this concept!