In the diagram, the circle will be dilated by a scale factor of 3 about the origin. The points *C*, *A*, and *B* map to *C'*, *A'*, and *B'* after the dilation. What is the length of C′⁢B′― ? Use the distance formula to help you decide: d=(x2−x1)2 + (y2−y1)2 . **A.** 5 units **B.** 15 units **C.** 21 units **D.** 24 units **E.** 45 units

To solve this problem, we can use the following steps:

1. **Understand the Dilation Process:**  
   The points \( C \), \( A \), and \( B \) are dilated by a scale factor of 3 about the origin. This means that the coordinates of each point are multiplied by 3.

2. **Determine the New Coordinates of \( C' \) and \( B' \):**  
   Let the original coordinates of \( C \) be \( (x_C, y_C) \) and the original coordinates of \( B \) be \( (x_B, y_B) \).  
   After dilation, the new coordinates \( C' \) and \( B' \) will be:
   \[
   C' = (3x_C, 3y_C)
   \]
   \[
   B' = (3x_B, 3y_B)
   \]

3. **Use the Distance Formula:**  
   The distance \( C'B' \) is given by the distance formula:
   \[
   d = \sqrt{(x_{B'} - x_{C'})^2 + (y_{B'} - y_{C'})^2}
   \]
   Substituting the coordinates of \( C' \) and \( B' \):
   \[
   d = \sqrt{(3x_B - 3x_C)^2 + (3y_B - 3y_C)^2}
   \]
   Simplify the expression:
   \[
   d = \sqrt{9[(x_B - x_C)^2 + (y_B - y_C)^2]}
   \]
   \[
   d = 3 \sqrt{(x_B - x_C)^2 + (y_B - y_C)^2}
   \]
   This means that the distance between \( C \) and \( B \) is scaled by a factor of 3.

4. **Conclusion:**  
   The original distance between points \( C \) and \( B \) before dilation must be 5 units, because after dilation it is scaled by 3, resulting in \( C'B' = 15 \) units. Therefore, the correct answer is:

**B. 15 units**

Would you like more details or have any questions? Here are some related questions for further exploration:

1. How do different scale factors affect the size and position of geometric figures after dilation?
2. What happens to the angles of a shape after it is dilated?
3. Can a dilation change the shape of a figure? Why or why not?
4. How does the dilation process differ when performed around a point other than the origin?
5. What is the impact of a dilation with a scale factor of less than 1?
6. How can you determine the coordinates of a point after dilation without using the origin as the center?
7. What other transformations, besides dilation, can affect the size of a geometric figure?
8. How does dilation relate to similarity in geometry?

**Tip:** When solving geometry problems involving transformations, sketching a diagram often helps to visualize the changes and verify your calculations.

Learn how to calculate the length of C′⁢B′ after dilation by a scale factor of 3 in this geometry problem. Use the distance formula to determine the distance between C′ and B′. Suitable for high school students studying geometry and transformations.

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