Question

F (-5,5)
H (-5,1)
G (-2,3)

J (5,-5)
K (9,-5)
I (9,-8)

Assess two figures using the definitions of transformations to determine whether the figures are similar. Is triangle FGH ~ IJK, and for what reason?

a. Yes. All sides are proportional
b. No. If point J was at (6,-6), the triangles would be similar
c. Yes. All angles are congruent
d. No. If point J was at (-6,6), the triangles would be similar

Answer 1

To determine whether triangle FGH is similar to triangle IJK, we can analyze their vertices and calculate the sides of both triangles.

### Points:
- Triangle FGH: F (-5,5), G (-2,3), H (-5,1)
- Triangle IJK: I (9,-8), J (5,-5), K (9,-5)

### Step 1: Calculate the lengths of the sides of triangle FGH
1. FG = sqrt[(-2 - (-5))^2 + (3 - 5)^2] = sqrt[(3)^2 + (-2)^2] = sqrt[9 + 4] = sqrt[13]
2. GH = sqrt[(-5 - (-2))^2 + (1 - 3)^2] = sqrt[(-3)^2 + (-2)^2] = sqrt[9 + 4] = sqrt[13]
3. HF = sqrt[(-5 - (-5))^2 + (5 - 1)^2] = sqrt[0^2 + (4)^2] = sqrt[16] = 4

So, the side lengths of triangle FGH are:
- FG = sqrt(13)
- GH = sqrt(13)
- HF = 4

### Step 2: Calculate the lengths of the sides of triangle IJK
1. IJ = sqrt[(5 - 9)^2 + (-5 - (-8))^2] = sqrt[(-4)^2 + (3)^2] = sqrt[16 + 9] = sqrt[25] = 5
2. JK = sqrt[(9 - 5)^2 + (-5 - (-5))^2] = sqrt[(4)^2 + 0^2] = sqrt[16] = 4
3. KI = sqrt[(9 - 9)^2 + (-8 - (-5))^2] = sqrt[0^2 + (-3)^2] = sqrt[9] = 3

So, the side lengths of triangle IJK are:
- IJ = 5
- JK = 4
- KI = 3

### Step 3: Assess similarity
To verify whether the triangles are similar, we need to check if either:
1. All corresponding side lengths are in proportion (side-side-side, SSS).
2. All corresponding angles are congruent (angle-angle, AA).

#### Side Lengths
The ratios of corresponding sides need to be compared:

- FG / IJ = sqrt(13) / 5
- GH / JK = sqrt(13) / 4
- HF / KI = 4 / 3

The ratios of corresponding sides are not equal, therefore the triangles are not similar based on the SSS criterion.

So, based on the analysis:

### Conclusion
The answer to whether triangle FGH is similar to triangle IJK is **b. No. If point J was at (6,-6), the triangles would be similar**. This suggests a reason related to the positioning of point J, but fundamentally, the triangles are not similar as per the calculations above.