Math Problem Statement

Triangle ABC has vertices A(2, -4), B(6, -2), and C(4, 8). After a dilation, the image has vertices A'(3, -6), B'(9, -3), and C'(6, 12). What scale factor was used to dilate ΔABC to ΔA'B'C'?

Solution

To solve this problem, we need to find the scale factor used to dilate triangle ΔABC\Delta ABC to triangle ΔABC\Delta A'B'C'.

Step 1: Identify the coordinates before and after dilation

  • Triangle ΔABC\Delta ABC has vertices:

    • A(2,4)A(2, -4)
    • B(6,2)B(6, -2)
    • C(4,8)C(4, 8)
  • After dilation, triangle ΔABC\Delta A'B'C' has vertices:

    • A(3,6)A'(3, -6)
    • B(9,3)B'(9, -3)
    • C(6,12)C'(6, 12)

Step 2: Use the formula for dilation

The formula for dilation is: (x,y)=(kx,ky)(x', y') = (kx, ky) where kk is the scale factor, and (x,y)(x, y) are the original coordinates.

Step 3: Calculate the scale factor for each point

  • From A(2,4)A(2, -4) to A(3,6)A'(3, -6): k=32=64=1.5k = \frac{3}{2} = \frac{-6}{-4} = 1.5

  • From B(6,2)B(6, -2) to B(9,3)B'(9, -3): k=96=32=1.5k = \frac{9}{6} = \frac{-3}{-2} = 1.5

  • From C(4,8)C(4, 8) to C(6,12)C'(6, 12): k=64=128=1.5k = \frac{6}{4} = \frac{12}{8} = 1.5

Step 4: Conclusion

The scale factor used to dilate ΔABC\Delta ABC to ΔABC\Delta A'B'C' is 1.5\boxed{1.5}.

Would you like further details or have any additional questions?

Here are five related questions to expand the topic:

  1. How would you determine the center of dilation?
  2. What happens if the scale factor is less than 1?
  3. Can you perform a dilation with a negative scale factor?
  4. How do you find the area of a dilated triangle compared to the original?
  5. How would the scale factor affect the distance between points?

Tip: When dilating, the scale factor applies equally to both the x and y coordinates, ensuring the proportional enlargement or reduction of the figure.

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Math Problem Analysis

Mathematical Concepts

Geometry
Transformations
Dilation

Formulas

(x', y') = (kx, ky)

Theorems

Dilation theorem

Suitable Grade Level

Grades 8-10