To determine if triangles ΔFGH and ΔIJK are similar, we need to check if their corresponding sides are proportional and/or if their corresponding angles are congruent.
- Calculate the lengths of the sides of ΔFGH:
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Length of FG: \[ FG = \sqrt{((-2) - (-5))^2 + (3 - 5)^2} = \sqrt{(3)^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13} \]
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Length of GH: \[ GH = \sqrt{((-5) - (-2))^2 + (5 - 3)^2} = \sqrt{(-3)^2 + (2)^2} = \sqrt{9 + 4} = \sqrt{13} \]
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Length of FH: \[ FH = \sqrt{((-5) - (-5))^2 + (5 - 5)^2} = \sqrt{(0)^2 + (0)^2} = 0 \]
- Calculate the lengths of the sides of ΔIJK:
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Length of IJ: \[ IJ = \sqrt{(4 - 9)^2 + (-5 - (-9))^2} = \sqrt{(-5)^2 + (4)^2} = \sqrt{25 + 16} = \sqrt{41} \]
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Length of JK: \[ JK = \sqrt{(9 - 4)^2 + (-4 - (-5))^2} = \sqrt{(5)^2 + (1)^2} = \sqrt{25 + 1} = \sqrt{26} \]
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Length of IK: \[ IK = \sqrt{(9 - 9)^2 + (-4 - (-9))^2} = \sqrt{(0)^2 + (5)^2} = 5 \]
- Check for proportionality:
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The side lengths for ΔFGH are:
- FG = \( \sqrt{13} \)
- GH = \( \sqrt{13} \)
- FH = \( 0 \)
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The side lengths for ΔIJK are:
- IJ = \( \sqrt{41} \)
- JK = \( \sqrt{26} \)
- IK = \( 5 \)
Since ΔFGH has a side length of 0 (indicating it collapses to a point), it does not have a valid shape and cannot be similar to any triangle. Furthermore, we could also check for angles, but that wouldn't change the conclusion because one of the triangles isn't a valid triangle.
Conclusion: ΔFGH is not similar to ΔIJK, therefore, the correct statement is: "No. if point J was at (6, -6), the triangles would be similar."