To determine the coordinates of the dilated trapezoid EFGH after a dilation with a scale factor of 3 about the origin, we can use the formula for dilation of a point \( (x, y) \) about the origin:
\[ (x, y) \rightarrow (kx, ky) \]
where \( k \) is the scale factor.
Given the vertices of trapezoid EFGH:
- E (1, 1)
- F (3, 1)
- G (4, -2)
- H (0, -2)
Applying the dilation with a scale factor of 3:
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Vertex E: \[ E' = (3 \cdot 1, 3 \cdot 1) = (3, 3) \]
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Vertex F: \[ F' = (3 \cdot 3, 3 \cdot 1) = (9, 3) \]
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Vertex G: \[ G' = (3 \cdot 4, 3 \cdot -2) = (12, -6) \]
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Vertex H: \[ H' = (3 \cdot 0, 3 \cdot -2) = (0, -6) \]
Now we have the new vertices of the dilated trapezoid E'F'G'H':
- E' (3, 3)
- F' (9, 3)
- G' (12, -6)
- H' (0, -6)
Statements about the vertices of image E'F'G'H':
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The lengths of the bases of the trapezoid E'F'G'H' are proportional to the original trapezoid EFGH.
- Since dilation maintains the shape but scales the distance, if EFGH had bases EF and GH, E'F' will be a scaled version of EF, and likewise for GH.
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The trapezoid E'F'G'H' is similar to trapezoid EFGH.
- All corresponding angles are preserved, and the ratios of corresponding sides remain constant, which is a property of similar figures.
Therefore, the two true statements about the vertices of image E'F'G'H' are that their corresponding sides are proportional and the trapezoids are similar.
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