To determine whether the triangles \( \Delta FGH \) and \( \Delta IJK \) are similar, we can use the definitions of transformations, specifically the properties of similar triangles:
- The angles of the triangles must be congruent.
- The sides of the triangles must be in proportion.
Given points for the triangles:
-
For \( \Delta FGH \):
- \( F(-5, 5) \)
- \( G(-2, 3) \)
- \( H(-5, 1) \)
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For \( \Delta IJK \):
- \( I(9, -8) \)
- \( J(-5, 4) \)
- \( K(9, -4) \)
Calculating side lengths:
We can use the distance formula to calculate the lengths of the sides of the triangles.
Side lengths of \( \Delta FGH \):
-
\( FG \): \[ FG = \sqrt{(-5 - (-2))^2 + (5 - 3)^2} = \sqrt{(-5 + 2)^2 + (5 - 3)^2} = \sqrt{(-3)^2 + (2)^2} = \sqrt{9 + 4} = \sqrt{13} \]
-
\( GH \): \[ GH = \sqrt{(-2 - (-5))^2 + (3 - 1)^2} = \sqrt{(-2 + 5)^2 + (2)^2} = \sqrt{(3)^2 + (2)^2} = \sqrt{9 + 4} = \sqrt{13} \]
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\( FH \): \[ FH = \sqrt{(-5 - (-5))^2 + (5 - 1)^2} = \sqrt{(0)^2 + (4)^2} = \sqrt{16} = 4 \]
Side lengths of \( \Delta IJK \):
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\( IJ \): \[ IJ = \sqrt{(9 - (-5))^2 + (-8 - 4)^2} = \sqrt{(9 + 5)^2 + (-8 - 4)^2} = \sqrt{(14)^2 + (-12)^2} = \sqrt{196 + 144} = \sqrt{340} \]
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\( JK \): \[ JK = \sqrt{(9 - (-5))^2 + (-4 - 4)^2} = \sqrt{(9 + 5)^2 + (-4 - 4)^2} = \sqrt{(14)^2 + (-8)^2} = \sqrt{196 + 64} = \sqrt{260} \]
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\( IK \): \[ IK = \sqrt{(9 - 9)^2 + (-8 - (-4))^2} = \sqrt{(0)^2 + (-8 + 4)^2} = \sqrt{0 + (-4)^2} = \sqrt{16} = 4 \]
Comparing side lengths:
-
For \( \Delta FGH \):
- \( FG = \sqrt{13} \)
- \( GH = \sqrt{13} \)
- \( FH = 4 \)
-
For \( \Delta IJK \):
- \( IJ = \sqrt{340} \)
- \( JK = \sqrt{260} \)
- \( IK = 4 \)
Proportionality and conclusion:
The sides of the triangles are not in proportion, and because their angles don't necessarily match up (we've only compared sides), we can conclude that the triangles are not similar based on the similarity criteria.
Final Assessment:
No, \( \Delta FGH \) and \( \Delta IJK \) are not similar; the correct response is: No. If point J was at (6, -6), the triangles would be similar.
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