https://file.aiquickdraw.com/mathbot/file/17294538385069u1h62n4.jpg

To solve this problem, we need to determine the scale factor and the center of enlargement for both transformations: from shape A to shape B and from shape B to shape A.

### Step 1: **Determine the Scale Factor (A to B)**
The shape B appears to be a smaller version of shape A. To find the scale factor, we can compare the lengths of corresponding sides of the two shapes.

- Shape A has a width of 2 units (from \(x = 3\) to \(x = 5\)) and a height of 6 units (from \(y = 1\) to \(y = 7\)).
- Shape B has a width of 1 unit (from \(x = 8\) to \(x = 9\)) and a height of 3 units (from \(y = 3\) to \(y = 6\)).

So, the scale factor from A to B is:
\[
\text{Scale factor from A to B} = \frac{\text{Side length of B}}{\text{Side length of A}} = \frac{1}{2} = 0.5
\]

### Step 2: **Find the Center of Enlargement (A to B)**
To find the center of enlargement, we need to trace back how the points on shape A map to shape B. This can be done by drawing lines through corresponding points on A and B, and finding the intersection of these lines.

- Line through top left of A (3, 7) and top left of B (8, 6).
- Line through bottom left of A (3, 1) and bottom left of B (8, 3).

These lines intersect at the point \((7, 1)\), which is the center of enlargement.

### Step 3: **Enlargement from B to A**
Since going from B to A is the reverse process, the scale factor is the reciprocal of the original. So, the scale factor from B to A is:
\[
\text{Scale factor from B to A} = \frac{1}{0.5} = 2
\]
The center of enlargement remains the same, at \((7, 1)\).

### Final Answers:
1. A to B: Enlargement with a scale factor of **0.5** and center **(7, 1)**.
2. B to A: Enlargement with a scale factor of **2** and center **(7, 1)**.

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. How do you calculate the scale factor between two shapes?
2. What happens to a shape’s orientation when it’s enlarged?
3. How does the location of the center of enlargement affect the transformation?
4. Can you describe what a negative scale factor would imply in an enlargement?
5. What other geometric transformations can you apply to shapes besides enlargement?

**Tip:** Always check the proportional dimensions of shapes when finding scale factors to avoid errors.

Complete the sentences below to fully describe the enlargements: A to B and B to A in the diagram.

This problem involves determining the scale factor and center of enlargement between two shapes, A and B. The scale factor from A to B is 0.5, and the center of enlargement is (7, 1). The reverse transformation from B to A has a scale factor of 2. Suitable for students in Grades 7-9.

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