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):

$\frac{5}{15} = \frac{1}{3}$

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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):

$\frac{4}{12} = \frac{1}{3}$

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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 $\frac{1}{3}$.

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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.