Question

Assess two figures using the definitions of transformations to determine whether the figures are similar. Is
ΔFGH∼ΔIJK
, and for what reason?

F(-5, 5) G(-2, 3) H(-5, 5)
I(9,-9) J(4, -5) K(9, -4)

yes, all sides are proportional

no. if point J was at (6, -6), the triangles would be similar

yes. all angles are congruent

no. if point J was at (-6,6), the triangles would be similar

Answer 1

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.

1. **Calculate the lengths of the sides of ΔFGH**:

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

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

- Length of FH:
\[
FH = \sqrt{((-5) - (-5))^2 + (5 - 5)^2} = \sqrt{(0)^2 + (0)^2} = 0
\]

2. **Calculate the lengths of the sides of ΔIJK**:

- Length of IJ:
\[
IJ = \sqrt{(4 - 9)^2 + (-5 - (-9))^2} = \sqrt{(-5)^2 + (4)^2} = \sqrt{25 + 16} = \sqrt{41}
\]

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

- Length of IK:
\[
IK = \sqrt{(9 - 9)^2 + (-4 - (-9))^2} = \sqrt{(0)^2 + (5)^2} = 5
\]

3. **Check for proportionality:**

- The side lengths for ΔFGH are:
- FG = \( \sqrt{13} \)
- GH = \( \sqrt{13} \)
- FH = \( 0 \)

- 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.**"