To determine if two rectangles (Figures A and B) are similar, we need to compare the ratios of their corresponding sides. If the ratios of corresponding sides are equal, the rectangles are similar.

Step-by-step Solution:

Figure A Dimensions:

• Horizontal side: 5 yards
• Vertical side: 4 yards

Figure B Dimensions:

• Horizontal side: 15 yards
• Vertical side: 12 yards

Let's find the ratios for the vertical and horizontal sides from Figure A to Figure B and vice versa.

1. Ratio: Vertical A to B

$\text{Vertical A to B (unsimplified)} = \frac{\text{Vertical side of A}}{\text{Vertical side of B}} = \frac{4}{12}$

$\text{Vertical A to B (simplified)} = \frac{4}{12} = \frac{1}{3}$

2. Ratio: Horizontal A to B

$\text{Horizontal A to B (unsimplified)} = \frac{\text{Horizontal side of A}}{\text{Horizontal side of B}} = \frac{5}{15}$

$\text{Horizontal A to B (simplified)} = \frac{5}{15} = \frac{1}{3}$

3. Ratio: Vertical B to A

$\text{Vertical B to A (unsimplified)} = \frac{\text{Vertical side of B}}{\text{Vertical side of A}} = \frac{12}{4}$

$\text{Vertical B to A (simplified)} = \frac{12}{4} = 3$

4. Ratio: Horizontal B to A

$\text{Horizontal B to A (unsimplified)} = \frac{\text{Horizontal side of B}}{\text{Horizontal side of A}} = \frac{15}{5}$

$\text{Horizontal B to A (simplified)} = \frac{15}{5} = 3$

Verification of Similarity

The ratios of corresponding sides from Figure A to Figure B are both $\frac{1}{3}$, and from Figure B to Figure A are both $3$. Since these ratios are consistent, the rectangles are indeed similar.

Scale Factors

• Scale factor from Figure A to Figure B: To scale Figure A to the size of Figure B, multiply each side length of Figure A by $\frac{1}{3}$.

$\text{Scale factor from A to B} = \frac{1}{3}$

• Scale factor from Figure B to Figure A: To scale Figure B to the size of Figure A, multiply each side length of Figure B by $3$.

$\text{Scale factor from B to A} = 3$

Summary

• Vertical A to B Ratio: Unsimplified: $\frac{4}{12}$, Simplified: $\frac{1}{3}$
• Horizontal A to B Ratio: Unsimplified: $\frac{5}{15}$, Simplified: $\frac{1}{3}$
• Vertical B to A Ratio: Unsimplified: $\frac{12}{4}$, Simplified: $3$
• Horizontal B to A Ratio: Unsimplified: $\frac{15}{5}$, Simplified: $3$

Would you like any more details or have any questions?

Related Questions

1. How do you find the scale factor if only one side length is given?
2. What happens if the ratios of corresponding sides are not equal?
3. How would you find the scale factor for figures that are not rectangles?
4. Can two figures with different shapes have a scale factor? Explain.
5. What are some practical applications of scale factors in real-world scenarios?

Tip: When dealing with scale factors, always ensure to compare corresponding sides to maintain consistency in calculations.