https://file.aiquickdraw.com/mathbot/file/1725481593534vpam3p50.jpg

To solve this problem, let's analyze the given information and perform the required calculations.

### Given:
1. **Triangle ABC** is being enlarged to form **triangle A'B'C'** using:
   - **Scale factor**: \( 3 \)
   - **Center of enlargement**: \( (1, 0) \)
   
2. We are asked to find:
   - (a) The coordinates of the vertex \( C' \).
   - (b) The length of side \( A'B' \).

### Solution:

#### Step 1: Identify Original Coordinates
From the graph, we can identify the coordinates of the original triangle \( ABC \):
- \( A = (1, 1) \)
- \( B = (3, 1) \)
- \( C = (2, 3) \)

#### Step 2: Calculate the Coordinates of Vertex C'
To find the coordinates of the enlarged vertex \( C' \), we use the formula for the coordinates of a point after enlargement:

\[
(x', y') = \left( \text{center}_x + \text{scale factor} \times (x - \text{center}_x), \; \text{center}_y + \text{scale factor} \times (y - \text{center}_y) \right)
\]

Here:
- **Center of enlargement**: \( (1, 0) \)
- **Scale factor**: \( 3 \)
- Original coordinates of \( C \): \( (2, 3) \)

Plugging in the values:

\[
x' = 1 + 3 \times (2 - 1) = 1 + 3 \times 1 = 1 + 3 = 4
\]

\[
y' = 0 + 3 \times (3 - 0) = 0 + 3 \times 3 = 0 + 9 = 9
\]

So, the coordinates of \( C' \) are \( (4, 9) \).

#### Step 3: Calculate the Length of Side A'B'
To find the length of side \( A'B' \), we first calculate the coordinates of \( A' \) and \( B' \).

**Original points:**
- \( A = (1, 1) \)
- \( B = (3, 1) \)

**Calculate new coordinates after enlargement:**

For \( A' \):
\[
x'_{A'} = 1 + 3 \times (1 - 1) = 1 + 3 \times 0 = 1
\]
\[
y'_{A'} = 0 + 3 \times (1 - 0) = 0 + 3 \times 1 = 3
\]
Thus, \( A' = (1, 3) \).

For \( B' \):
\[
x'_{B'} = 1 + 3 \times (3 - 1) = 1 + 3 \times 2 = 1 + 6 = 7
\]
\[
y'_{B'} = 0 + 3 \times (1 - 0) = 0 + 3 \times 1 = 3
\]
Thus, \( B' = (7, 3) \).

**Length of \( A'B' \):**
The formula for the distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

\[
\text{Length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Substitute the coordinates of \( A' = (1, 3) \) and \( B' = (7, 3) \):

\[
\text{Length of } A'B' = \sqrt{(7 - 1)^2 + (3 - 3)^2} = \sqrt{6^2 + 0} = \sqrt{36} = 6
\]

### Final Answers:
(a) The coordinates of the vertex \( C' \) are \( (4, 9) \).

(b) The length of side \( A'B' \) is \( 6 \) units.

Would you like further details or have any questions?

### Related Questions:
1. How do you find the coordinates of other points after an enlargement with a different scale factor?
2. What happens if the center of enlargement is outside the triangle?
3. Can we find the area of the enlarged triangle \( A'B'C' \) and compare it with the original triangle?
4. How does changing the center of enlargement affect the new coordinates of the vertices?
5. What would be the new coordinates of \( A'B'C' \) if the scale factor was changed to \( 2 \)?

#### Tip:
When enlarging or reducing a shape, always use the center of enlargement as your reference point to calculate new positions accurately.

Triangle ABC is being enlarged using a scale factor of 3 and center (1, 0) to give triangle A'B'C'.
(a) What are the coordinates of the vertex C'?
(b) What is the length of the side A'B'?

Learn how to find the coordinates of a vertex after enlargement and calculate the side length of the enlarged triangle using the scale factor and the center of enlargement. This problem involves coordinate geometry, enlargement formulas, and the distance formula, suitable for students in Grades 8-10.

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