To determine the transformations that produced triangle \( \triangle ABC \) from triangle \( \triangle XYZ \) and to assess their congruence, we need to analyze the coordinates of each triangle.
Vertices of Triangle ABC:
- \( A(-5, 2) \)
- \( B(-3, 2) \)
- \( C(-3, 5) \)
Vertices of Triangle XYZ:
- \( Z(-3, -4) \)
- \( Y(-4, 0) \)
- \( X(-6, 0) \)
Step 1: Find the side lengths of both triangles
Calculating the side lengths of Triangle ABC:
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Length of side \( AB \): \[ AB = \sqrt{((-5) - (-3))^2 + (2 - 2)^2} = \sqrt{(-2)^2} = 2 \]
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Length of side \( BC \): \[ BC = \sqrt{((-3) - (-3))^2 + (2 - 5)^2} = \sqrt{0^2 + (-3)^2} = 3 \]
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Length of side \( CA \): \[ CA = \sqrt{((-5) - (-3))^2 + (2 - 5)^2} = \sqrt{(-2)^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13} \]
The side lengths of triangle ABC are:
- AB = 2
- BC = 3
- CA = \( \sqrt{13} \)
Calculating the side lengths of Triangle XYZ:
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Length of side \( YZ \): \[ YZ = \sqrt{((-4) - (-3))^2 + (0 - (-4))^2} = \sqrt{(-1)^2 + (4)^2} = \sqrt{1 + 16} = \sqrt{17} \]
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Length of side \( ZX \): \[ ZX = \sqrt{((-6) - (-3))^2 + (0 - (-4))^2} = \sqrt{(-3)^2 + (4)^2} = \sqrt{9 + 16} = 5 \]
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Length of side \( XY \): \[ XY = \sqrt{((-6) - (-4))^2 + (0 - 0)^2} = \sqrt{(-2)^2} = 2 \]
The side lengths of triangle XYZ are:
- YZ = \( \sqrt{17} \)
- ZX = 5
- XY = 2
Step 2: Compare the side lengths of both triangles
The side lengths of triangle ABC are \( (2, 3, \sqrt{13}) \) and the side lengths of triangle XYZ are \( (2, 5, \sqrt{17}) \).
Step 3: Congruence check
For triangles to be congruent, their corresponding sides must be equal in length.
Comparing side lengths:
- Side \( AB \) (2) corresponds to Side \( XY \) (2) - Equal
- Side \( BC \) (3) does not correspond to any equal side in \( \triangle XYZ \) - Not equal
- Side \( CA \) ( \( \sqrt{13} \) ) does not correspond to any equal side in \( \triangle XYZ \) - Not equal
Conclusion: Congruency assessment
Since not all corresponding sides are equal, the triangles \( \triangle ABC \) and \( \triangle XYZ \) are not congruent.
Assessing the transformations:
Given that the points of triangle \( \triangle XYZ \) appear to be different from those of triangle \( \triangle ABC \) in both orientation and distance, it indicates that a combination of transformations (such as translation, rotation, or reflection) was performed, but the triangles are not congruent since the lengths of at least one side do not match. Thus, we cannot outline specific transformations because congruency cannot be established.
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