Triangle FGH coordinates are F(-5,5), G(-2,3), H(-5,1)

Question

Triangle IJK coordinates are I(9,-8), J(4,-5), K(9,-4)
Assess two figures using the definitions of transformations to determine whether the figures are similar. is triangle FGH ~ triangle IJK, and for what reason?
A. Yes. All sides are proportional
B. Yes. all angles are congruent
C. No. if point J was at (-6,6), the triangles would be similar
D. No. if point J was at (6,-6), the triangles would be similar

Answer 1

To assess whether triangles FGH and IJK are similar, we can check for proportionality of sides and congruence of corresponding angles.

First, we can calculate the lengths of the sides for triangle FGH using the distance formula, which is defined as:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Calculating the lengths of the sides of triangle FGH:

1. FG: \[ FG = \sqrt{((-2) - (-5))^2 + (3 - 5)^2} = \sqrt{(3)^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13} \]

2. GH: \[ GH = \sqrt{((-5) - (-2))^2 + (1 - 3)^2} = \sqrt{(-3)^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13} \]

3. HF: \[ HF = \sqrt{((-5) - (-5))^2 + (5 - 1)^2} = \sqrt{(0)^2 + (4)^2} = \sqrt{16} = 4 \]

Thus, the side lengths of triangle FGH are:

• FG = \(\sqrt{13}\)
• GH = \(\sqrt{13}\)
• HF = 4

Calculating the lengths of the sides of triangle IJK:

1. IJ: \[ IJ = \sqrt{(4 - 9)^2 + (-5 - (-8))^2} = \sqrt{(-5)^2 + (3)^2} = \sqrt{25 + 9} = \sqrt{34} \]

2. JK: \[ JK = \sqrt{(9 - 4)^2 + (-4 - (-5))^2} = \sqrt{(5)^2 + (1)^2} = \sqrt{25 + 1} = \sqrt{26} \]

3. KI: \[ KI = \sqrt{(9 - 9)^2 + (-8 - (-4))^2} = \sqrt{(0)^2 + (-4)^2} = \sqrt{16} = 4 \]

Thus, the side lengths of triangle IJK are:

• IJ = \(\sqrt{34}\)
• JK = \(\sqrt{26}\)
• KI = 4

Checking for Similarity

For similarity, we should have:

1. Corresponding angles should be equal (which we cannot find without additional information).
2. The ratios of corresponding sides should be equal.

Let's check the side ratios:

• Side lengths in triangle FGH: \( \sqrt{13}, \sqrt{13}, 4 \)
• Side lengths in triangle IJK: \( \sqrt{34}, \sqrt{26}, 4 \)

The side with length 4 is the same in both triangles, but the other two sides are different. For the triangles to be similar, all corresponding ratios must be the same. Let's compare the ratios:

• Ratio of corresponding sides \( FG / IJ \): \[ \frac{\sqrt{13}}{\sqrt{34}} \quad (not ; equal) \]

• Ratio of corresponding sides \( GH / JK \): \[ \frac{\sqrt{13}}{\sqrt{26}} \quad (not ; equal) \]

Since the sides are not proportional, the triangles FGH and IJK are not similar.

Conclusion:

The answer is D. No. if point J was at (6, -6), the triangles would be similar.

This implies that if triangle IJK's coordinates were adjusted such that corresponding sides and angles matched with triangle FGH, similarity could potentially be established.