To find the scale factor needed to transform rectangle ABCD into rectangle EFGH using the given corner points, we first need to calculate the side lengths of both rectangles.
Rectangle ABCD:
- Points:
- A(-10, 6)
- B(-10, -2)
- C(-6, -2)
- D(-6, 6)
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Calculate the lengths of the sides of rectangle ABCD:
- Length AB (vertical side) = |y-coordinate A - y-coordinate B| = |6 - (-2)| = |6 + 2| = 8
- Length AD (horizontal side) = |x-coordinate A - x-coordinate D| = |-10 - (-6)| = |-10 + 6| = 4
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Side lengths of rectangle ABCD:
- Length = 8 (vertical), Width = 4 (horizontal)
Rectangle EFGH:
- Points:
- E(-5, 3)
- F(-5, -1)
- G(-3, -1)
- H(-3, 3)
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Calculate the lengths of the sides of rectangle EFGH:
- Length EF (vertical side) = |y-coordinate E - y-coordinate F| = |3 - (-1)| = |3 + 1| = 4
- Length EH (horizontal side) = |x-coordinate E - x-coordinate H| = |-5 - (-3)| = |-5 + 3| = 2
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Side lengths of rectangle EFGH:
- Length = 4 (vertical), Width = 2 (horizontal)
Finding the Scale Factor: The scale factor can be determined by comparing the corresponding side lengths of the two rectangles:
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For the vertical sides: \[ \text{Scale factor} = \frac{\text{Length of EFGH}}{\text{Length of ABCD}} = \frac{4}{8} = \frac{1}{2} \]
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For the horizontal sides: \[ \text{Scale factor} = \frac{\text{Width of EFGH}}{\text{Width of ABCD}} = \frac{2}{4} = \frac{1}{2} \]
Since both calculations yield the same scale factor, the transformation scale factor from rectangle ABCD to rectangle EFGH is \(\frac{1}{2}\).
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