The rectangle K'L'M'N' is a dilation of the rectangle KLNM. What is the scale factor of the dilation? Simplify your answer and write it as a proper fraction, an improper fraction,... The rectangle K'L'M'N' is a dilation of the rectangle KLNM. What is the scale factor of the dilation? Simplify your answer and write it as a proper fraction, an improper fraction, or a whole number.
Understand the Problem
The question asks to determine the scale factor of the dilation from rectangle KLNM to rectangle K'L'M'N'. This involves finding the ratio of the lengths of corresponding sides of the two rectangles.
Answer
The scale factor of the dilation is $ \frac{3}{4} $.
Answer for screen readers
The scale factor of the dilation is $ \frac{3}{4} $.
Steps to Solve
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Identify the dimensions of rectangle KLNM First, we need to identify the coordinates of the corners of rectangle KLNM.
- K: (-6, 6)
- L: (-6, 2)
- M: (-2, 2)
- N: (-2, 6)
The width can be calculated as the difference between the long sides (K and N): $$ \text{Width of KLNM} = |-2 - (-6)| = |4| = 4 $$
The height can be calculated as the difference between the short sides (K and L): $$ \text{Height of KLNM} = |6 - 2| = |4| = 4 $$
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Identify the dimensions of rectangle K'L'M'N' Now, we find the coordinates of rectangle K'L'M'N':
- K': (-4, 4)
- L': (-4, 1)
- M': (0, 1)
- N': (0, 4)
The width is: $$ \text{Width of K'L'M'N'} = |0 - (-4)| = |4| = 4 $$
The height is: $$ \text{Height of K'L'M'N'} = |4 - 1| = |3| = 3 $$
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Calculate the scale factors The scale factor can be calculated using the ratio of the lengths of corresponding sides:
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Scale factor for width: $$ \text{Scale Factor (Width)} = \frac{\text{Width of K'L'M'N'}}{\text{Width of KLNM}} = \frac{4}{4} = 1 $$
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Scale factor for height: $$ \text{Scale Factor (Height)} = \frac{\text{Height of K'L'M'N'}}{\text{Height of KLNM}} = \frac{3}{4} $$
We can conclude that the overall scale factor is determined by either dimension. In this case, we take the height scale factor as our main scale factor.
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The scale factor of the dilation is $ \frac{3}{4} $.
More Information
The scale factor compares the size of the two rectangles, indicating how much the rectangle KLNM has been scaled down to form rectangle K'L'M'N'. The factor $ \frac{3}{4} $ means that K'L'M'N' is 75% of the size of KLNM.
Tips
- Not identifying the corresponding sides correctly.
- Forgetting to check whether the rectangles are enlargements or reductions.
- Confusing width and height when calculating ratios.
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