To determine the scale factor of a dilation, we compare the coordinates of the original triangle with the dilated triangle. Here's the step-by-step process for this problem:

Given:

• Original triangle:
  • $A(0, 0), B(0, 4), C(6, 0)$
• Dilated triangle:
  • $A'(0, 0), B'(0, 10), C'(15, 0)$

Step 1: Understanding the scale factor

The scale factor is the ratio between corresponding distances in the dilated and original figures. Since point $A$ remains the same, we focus on the other coordinates.

1. Compare points $B$ and $B'$:

   • Original $B$: $(0, 4)$
   • Dilated $B'$: $(0, 10)$
   • Scale factor:
     $k = \frac{\text{New length}}{\text{Original length}} = \frac{10}{4} = 2.5$

2. Confirm with points $C$ and $C'$:

   • Original $C$: $(6, 0)$
   • Dilated $C'$: $(15, 0)$
   • Scale factor:
     $k = \frac{15}{6} = 2.5$

Step 2: Conclusion

The scale factor for this dilation is 2.5.

Correct Answer:

B) 2.5

Would you like me to explain any part further? Here are some related questions to explore:

1. What is the general formula for a dilation transformation?
2. How does the center of dilation affect the scale factor?
3. Can a dilation result in a reduction instead of an enlargement?
4. How do you determine the scale factor for 3D figures?
5. What happens to a figure's area when it undergoes a dilation?

Tip: Always verify the scale factor using multiple corresponding points to ensure accuracy.