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 to triangle .
Step 1: Identify the coordinates before and after dilation
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Triangle has vertices:
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After dilation, triangle has vertices:
Step 2: Use the formula for dilation
The formula for dilation is: where is the scale factor, and are the original coordinates.
Step 3: Calculate the scale factor for each point
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From to :
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From to :
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From to :
Step 4: Conclusion
The scale factor used to dilate to is .
Would you like further details or have any additional questions?
Here are five related questions to expand the topic:
- How would you determine the center of dilation?
- What happens if the scale factor is less than 1?
- Can you perform a dilation with a negative scale factor?
- How do you find the area of a dilated triangle compared to the original?
- 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