Question

Triangle FGH
Point F: (-5, 5)
Point G: (-2, 3)
Point H: (1, -5)

Triangle IJK:
Point I: (9, -8)
Point J: (4, -5)
Point 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. No. if point J was at (6, -6), the triangles would be similar.
C. No. If point J was at (-6, 6), the triangles would be similar.
D. Yes. All angles are congruent.

Answer 1

To determine if Triangle FGH is similar to Triangle IJK, we can use the definitions of transformation which include checking for proportional sides and congruent angles.

1. **Calculate the lengths of the sides of Triangle FGH:**
- FG:
\[
FG = \sqrt{(-2 - (-5))^2 + (3 - 5)^2} = \sqrt{(-2 + 5)^2 + (-2)^2} = \sqrt{3^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13}
\]

- GH:
\[
GH = \sqrt{(1 - (-2))^2 + (-5 - 3)^2} = \sqrt{(1 + 2)^2 + (-8)^2} = \sqrt{3^2 + 64} = \sqrt{9 + 64} = \sqrt{73}
\]

- FH:
\[
FH = \sqrt{(1 - (-5))^2 + (-5 - 5)^2} = \sqrt{(6)^2 + (-10)^2} = \sqrt{36 + 100} = \sqrt{136} = 2\sqrt{34}
\]

2. **Calculate the lengths of the sides of Triangle IJK:**
- IJ:
\[
IJ = \sqrt{(4 - 9)^2 + (-5 - (-8))^2} = \sqrt{(-5)^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34}
\]

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

- IK:
\[
IK = \sqrt{(9 - 9)^2 + (-4 - (-8))^2} = \sqrt{0 + 4^2} = 4
\]

3. **Proportionality check:**
By comparing the sides:
- The lengths from Triangle FGH:
\[
\sqrt{13}, \sqrt{73}, 2\sqrt{34}
\]
- The lengths from Triangle IJK:
\[
\sqrt{34}, \sqrt{26}, 4
\]
- Since there is no consistent ratio among the pairs of sides, the triangles do not have proportional sides.

4. **Angle Congruence check:**
We can also assess similarity through angle relationships, typically from the slopes of the lines formed by their vertices. However, this would require further calculation of angles.

Given that there is no evidence for all sides being proportional and no explicit indication of angle congruence, we can conclude:

**Answer: B. No. if point J was at (6, -6), the triangles would be similar.**

This suggests that modifying the position of the point in Triangle IJK would allow for the potential of similarity via angles or proportional sides.