To find the scale factor needed to transform rectangle EFGH to rectangle ABCD, we first need to determine the side lengths of both rectangles.
Finding the side lengths of rectangle ABCD:
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The length of rectangle ABCD can be calculated using the coordinates of vertices A and D (or A and B).
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The distance between A and D: \[ \text{Length} = |y_A - y_D| = |6 - 6| = 0 \quad (\text{this is incorrect as A and D lie on the same side, we mean} |-10 - (-6)|) \] Since AD is horizontal (points A and D have same y-coordinate): \[ \text{Length} = |-10 - (-6)| = |-10 + 6| = 4 \]
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The width of rectangle ABCD can be calculated using the coordinates of vertices A and B or B and C.
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The distance between A and B (or B and C): \[ \text{Width} = |y_A - y_B| = |6 - (-2)| = |6 + 2| = 8 \]
So, the dimensions of rectangle ABCD are:
- Length = 4
- Width = 8
Finding the side lengths of rectangle EFGH:
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The length of rectangle EFGH can similarly be calculated using points E and G and E and F.
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The distance between E and G (horizontal): \[ \text{Length} = |-5 - (-3)| = |-5 + 3| = 2 \]
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The width can be measured between E and F (vertical): \[ \text{Width} = |y_E - y_F| = |3 - (-1)| = |3 + 1| = 4 \]
So, the dimensions of rectangle EFGH are:
- Length = 2
- Width = 4
Ratio Calculation:
Now, we can find the scale factor from rectangle EFGH to rectangle ABCD:
- The scale factor for length: \[ \text{Scale Factor}_{length} = \frac{\text{Length of ABCD}}{\text{Length of EFGH}} = \frac{4}{2} = 2 \]
- The scale factor for width: \[ \text{Scale Factor}_{width} = \frac{\text{Width of ABCD}}{\text{Width of EFGH}} = \frac{8}{4} = 2 \]
Since both scale factors are consistent, the overall scale factor needed to convert from rectangle EFGH to rectangle ABCD is:
\[ \text{Scale Factor} = 2 \]
Thus, the scale factor needed to transform from EFGH to ABCD is 2.