To find the pre-image coordinates of trapezoid ABCD after applying the translation rule \( T_{-1, -3}(x, y) \), we need to reverse the translation. The translation rule \( T_{-1, -3}(x, y) \) means that every point (x, y) in the pre-image is transformed to (x - 1, y - 3) in the image.
Therefore, to find the pre-image coordinates A, B, C, and D from the image coordinates A′, B′, C′, and D′, we can apply the following transformations:
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For point A′(–2, –5): \[ A(-1, -2) = A'(-2 + 1, -5 + 3) = (-1, -2) \]
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For point B′(–1, –2): \[ B(0, 1) = B'(-1 + 1, -2 + 3) = (0, 1) \]
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For point C′(2, –2): \[ C(3, 1) = C'(2 + 1, -2 + 3) = (3, 1) \]
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For point D′(3, –5): \[ D(4, -2) = D'(3 + 1, -5 + 3) = (4, -2) \]
Thus, the pre-image coordinates are:
- A(–1, –2)
- B(0, 1)
- C(3, 1)
- D(4, –2)
Now, let’s check which of the given trapezoids corresponds to these points:
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First option: A(–1, –2), B(0, 1), C(3, 1), D(4, –2) — This matches our pre-image.
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Second option: A(–3, –2), B(–2, 1), C(1, 1), D(2, –2) — This does not match.
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Third option: A(–4, –5), B(–3, –2), C(0, –2), D(1, –5) — This does not match.
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Fourth option: A(0, –2), B(1, 1), C(4, 1), D(5, –2) — This does not match.
Therefore, the correct diagram that shows the pre-image of trapezoid ABCD is the first option:
- A(–1, –2), B(0, 1), C(3, 1), D(4, –2).