Question
How to find the scale factor and missing side of similar rectangles
Answers
GPT-4o mini
To find the scale factor and the missing side of similar rectangles, follow these steps:
1. **Identify Corresponding Sides**: First, you need to identify the lengths of corresponding sides of the similar rectangles. For example, if you have Rectangle A and Rectangle B, and you know two sides of Rectangle A (let's say \( a_1 \) and \( b_1 \)) and one side of Rectangle B (let's say \( b_2 \)), you can use these to find the scale factor.
2. **Calculate the Scale Factor**: The scale factor can be calculated by taking the ratio of the lengths of corresponding sides. If \( a_1 \) corresponds to \( a_2 \) (the missing side in Rectangle B), the scale factor (k) can be calculated as follows:
\[
k = \frac{b_2}{b_1}
\]
or
\[
k = \frac{a_2}{a_1}
\]
It’s important to use lengths from the same corresponding sides (i.e., both from Rectangle A or both from Rectangle B).
3. **Finding the Missing Side**: If you know the scale factor and one side of the similar rectangle, you can find the missing side using the relationship established by the scale factor. For example, if you have the scale factor \( k \) and you need to find \( a_2 \):
\[
a_2 = k \times a_1
\]
or conversely, if you have a side of Rectangle B and you want to determine the corresponding side in Rectangle A, the formula would be:
\[
a_1 = \frac{a_2}{k}
\]
### Example Problem:
Suppose you have Rectangle A with dimensions 4 cm by 6 cm, and you have Rectangle B with one side measuring 8 cm (let’s say it corresponds to the longer side of Rectangle A).
1. **Identify Corresponding Sides**: Longer side of Rectangle A is 6 cm and shorter side is 4 cm. The longer side of Rectangle B is 8 cm.
2. **Calculate the Scale Factor**:
\[
k = \frac{\text{longer side of Rectangle B}}{\text{longer side of Rectangle A}} = \frac{8}{6} = \frac{4}{3}
\]
3. **Finding the Missing Side**: Now we can find the shorter side of Rectangle B using the scale factor:
\[
\text{shorter side of Rectangle B} = k \times \text{shorter side of Rectangle A} = \frac{4}{3} \times 4 = \frac{16}{3} \approx 5.33 \text{ cm}
\]
So the dimensions of Rectangle B are 8 cm by approximately 5.33 cm.
1. **Identify Corresponding Sides**: First, you need to identify the lengths of corresponding sides of the similar rectangles. For example, if you have Rectangle A and Rectangle B, and you know two sides of Rectangle A (let's say \( a_1 \) and \( b_1 \)) and one side of Rectangle B (let's say \( b_2 \)), you can use these to find the scale factor.
2. **Calculate the Scale Factor**: The scale factor can be calculated by taking the ratio of the lengths of corresponding sides. If \( a_1 \) corresponds to \( a_2 \) (the missing side in Rectangle B), the scale factor (k) can be calculated as follows:
\[
k = \frac{b_2}{b_1}
\]
or
\[
k = \frac{a_2}{a_1}
\]
It’s important to use lengths from the same corresponding sides (i.e., both from Rectangle A or both from Rectangle B).
3. **Finding the Missing Side**: If you know the scale factor and one side of the similar rectangle, you can find the missing side using the relationship established by the scale factor. For example, if you have the scale factor \( k \) and you need to find \( a_2 \):
\[
a_2 = k \times a_1
\]
or conversely, if you have a side of Rectangle B and you want to determine the corresponding side in Rectangle A, the formula would be:
\[
a_1 = \frac{a_2}{k}
\]
### Example Problem:
Suppose you have Rectangle A with dimensions 4 cm by 6 cm, and you have Rectangle B with one side measuring 8 cm (let’s say it corresponds to the longer side of Rectangle A).
1. **Identify Corresponding Sides**: Longer side of Rectangle A is 6 cm and shorter side is 4 cm. The longer side of Rectangle B is 8 cm.
2. **Calculate the Scale Factor**:
\[
k = \frac{\text{longer side of Rectangle B}}{\text{longer side of Rectangle A}} = \frac{8}{6} = \frac{4}{3}
\]
3. **Finding the Missing Side**: Now we can find the shorter side of Rectangle B using the scale factor:
\[
\text{shorter side of Rectangle B} = k \times \text{shorter side of Rectangle A} = \frac{4}{3} \times 4 = \frac{16}{3} \approx 5.33 \text{ cm}
\]
So the dimensions of Rectangle B are 8 cm by approximately 5.33 cm.