Two figures are similar if they have the exact same shape and their corresponding sides are proportional. The corresponding side lengths of the two figures are related by a scale factor.
A scale factor is the constant number you can multiply any side length in one figure by to find the corresponding side length of the similar figure.
Figure A
5 yd. horizontal
4 yd. vertical
Figure B
15 yd. horizontal
12 yd. vertical
In this problem, you will verify that the two rectangles are similar and find the scale factor from each figure to the other. Since rectangles have two sets of equal parallel sides, we will only need to find the ratios for two sets of sides; the horizontal and vertical lengths. If these ratios are equal, the rectangles are similar.

Note: In some cases, the simplified and unsimplifed ratios may be equal.
Find the ratio:

Unsimplified:

Simplified:

Find the ratio:

Unsimplified:

Simplified:

Find the ratio:

Unsimplified:

Simplified:

Find the ratio:

Unsimplified:

Simplified:

To scale Figure A to the size of Figure B, multiply the length of each side of Figure A by the scale factor of

To scale Figure B to the size of Figure A, multiply the length of each side of Figure B by the scale factor of

We are given two rectangles, **Figure A** and **Figure B**, and we need to verify their similarity by comparing the ratios of their corresponding side lengths (horizontal and vertical). If these ratios are equal, the two figures are similar. Additionally, we will find the scale factors to scale one figure to the size of the other.

### Step 1: Calculate the ratios for the horizontal sides

The horizontal side lengths are:
- **Figure A**: 5 yd
- **Figure B**: 15 yd

#### Unsimplified ratio (Figure A to Figure B):
\[
\frac{5}{15} = \frac{5}{15}
\]

#### Simplified ratio (Figure A to Figure B):
\[
\frac{5}{15} = \frac{1}{3}
\]

---

### Step 2: Calculate the ratios for the vertical sides

The vertical side lengths are:
- **Figure A**: 4 yd
- **Figure B**: 12 yd

#### Unsimplified ratio (Figure A to Figure B):
\[
\frac{4}{12} = \frac{4}{12}
\]

#### Simplified ratio (Figure A to Figure B):
\[
\frac{4}{12} = \frac{1}{3}
\]

---

### Step 3: Verify the similarity of the rectangles

Since both the horizontal and vertical ratios (unsimplified and simplified) are equal, the two rectangles are **similar**. The scale factor is the same for both dimensions, which confirms the similarity.

### Step 4: Find the scale factors

#### Scale factor to scale **Figure A** to the size of **Figure B**:
The scale factor is:
\[
\frac{\text{Length of side in Figure B}}{\text{Length of corresponding side in Figure A}} = \frac{15}{5} = 3
\]
Thus, to scale **Figure A** to the size of **Figure B**, multiply each side of **Figure A** by **3**.

#### Scale factor to scale **Figure B** to the size of **Figure A**:
The scale factor is:
\[
\frac{\text{Length of side in Figure A}}{\text{Length of corresponding side in Figure B}} = \frac{5}{15} = \frac{1}{3}
\]
Thus, to scale **Figure B** to the size of **Figure A**, multiply each side of **Figure B** by **\(\frac{1}{3}\)**.

---

### Summary:

- **Horizontal ratio (unsimplified)**: \(\frac{5}{15}\)
- **Horizontal ratio (simplified)**: \(\frac{1}{3}\)
- **Vertical ratio (unsimplified)**: \(\frac{4}{12}\)
- **Vertical ratio (simplified)**: \(\frac{1}{3}\)
- **Scale factor (Figure A to B)**: \(3\)
- **Scale factor (Figure B to A)**: \(\frac{1}{3}\)

Would you like more details on any step or have any other questions?

### Related Questions:
1. How do you find the scale factor between two similar figures?
2. What is the difference between similar and congruent figures?
3. Can you scale a 3D figure in the same way as a 2D figure?
4. How does the scale factor affect the area of similar figures?
5. How do you verify that two triangles are similar using side ratios?

### Tip:
When working with ratios and scale factors, always simplify the ratios to ensure the correct comparison between figures.

Two figures are similar if they have the exact same shape and their corresponding sides are proportional. The corresponding side lengths of the two figures are related by a scale factor. Figure A 5 yd. horizontal 4 yd. vertical Figure B 15 yd. horizontal 12 yd. vertical. Verify that the two rectangles are similar and find the scale factor from each figure to the other.

Explore how to verify the similarity of two rectangles, Figure A and Figure B, using ratios and scale factors. This problem includes calculating the horizontal and vertical side ratios, simplifying them, and determining the scale factors. Suitable for students in Grades 6-8.

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