Question

Step 1: Calculate the side lengths of triangle ABC. Using the distance formula, d=(x2−x1)2+(y2−y1)2:

AB: (−3−(−5))2+(2−2)2=(2)2+(0)2=4=2 units [1]
BC: (−3−(−3))2+(5−2)2=(0)2+(3)2=9=3 units [1]
AC: (−3−(−5))2+(5−2)2=(2)2+(3)2=4+9=13 units [1]
Step 2: Calculate the side lengths of triangle XYZ.

XY: (0−0)2+(−4−(−6))2=(0)2+(2)2=4=2 units [2]
YZ: (−3−0)2+(4−(−4))2=(−3)2+(8)2=9+64=73 units [2]
XZ: (−3−0)2+(4−(−6))2=(−3)2+(10)2=9+100=109 units [2]
Step 3: Compare the side lengths.

AB = 2, XY = 2
BC = 3, YZ = 73
AC = 13, XZ = 109
Since not all corresponding side lengths are equal (e.g., BC ≠ YZ and AC ≠ XZ), the two triangles are not congruent. [3]

Because the triangles are not congruent, a simple sequence of rigid transformations (translation, rotation, reflection) cannot transform triangle ABC into triangle XYZ. If they were congruent, we would look for a combination of these transformations. However, since their side lengths differ significantly, it indicates that a dilation (a non-rigid transformation that changes size) would be involved, or they are simply not related by a direct sequence of transformations that preserve size and shape. [4]

The transformations made to ABC to produce XYZ are not a sequence of rigid transformations (translation, rotation, reflection) because the triangles are not congruent. [3]

How do you know? Show your work.
We know this because we calculated the lengths of all sides for both triangles. For two triangles to be congruent, all corresponding sides must have equal lengths (SSS Congruence Postulate), or other congruence postulates like SAS, ASA, AAS, or HL must apply. In this case, comparing the side lengths:

AB = 2 and XY = 2 (These are equal)
BC = 3 and YZ = 73 (These are not equal, as 73≈8.54)
AC = 13 and XZ = 109 (These are not equal, as 13≈3.61 and 109≈10.44)

Answer 1

Let's summarize and verify the calculations step-by-step for clarity:

### Step 1: Calculate the side lengths of triangle ABC

1. **Length of AB:**
\[
d_{AB} = \sqrt{(-3 - (-5))^2 + (2 - 2)^2} = \sqrt{(2)^2 + (0)^2} = \sqrt{4} = 2 \text{ units}
\]

2. **Length of BC:**
\[
d_{BC} = \sqrt{(-3 - (-3))^2 + (5 - 2)^2} = \sqrt{(0)^2 + (3)^2} = \sqrt{9} = 3 \text{ units}
\]

3. **Length of AC:**
\[
d_{AC} = \sqrt{(-3 - (-5))^2 + (5 - 2)^2} = \sqrt{(2)^2 + (3)^2} = \sqrt{4 + 9} = \sqrt{13} \text{ units}
\]

### Step 2: Calculate the side lengths of triangle XYZ

1. **Length of XY:**
\[
d_{XY} = \sqrt{(0 - 0)^2 + (-4 - (-6))^2} = \sqrt{(0)^2 + (2)^2} = \sqrt{4} = 2 \text{ units}
\]

2. **Length of YZ:**
\[
d_{YZ} = \sqrt{(-3 - 0)^2 + (4 - (-4))^2} = \sqrt{(-3)^2 + (8)^2} = \sqrt{9 + 64} = \sqrt{73} \text{ units}
\]

3. **Length of XZ:**
\[
d_{XZ} = \sqrt{(-3 - 0)^2 + (4 - (-6))^2} = \sqrt{(-3)^2 + (10)^2} = \sqrt{9 + 100} = \sqrt{109} \text{ units}
\]

### Step 3: Compare the side lengths

Now let’s compare the lengths:

- **AB**: 2 units, **XY**: 2 units → Equal
- **BC**: 3 units, **YZ**: \(\sqrt{73} \approx 8.54\) units → Not equal
- **AC**: \(\sqrt{13} \approx 3.61\) units, **XZ**: \(\sqrt{109} \approx 10.44\) units → Not equal

### Conclusion

Since not all corresponding side lengths are equal (only AB = XY), triangle ABC and triangle XYZ are **not congruent**. The requirement for congruence is that all corresponding sides must be equal.

Because triangle ABC and triangle XYZ are not congruent, rigid transformations (translation, rotation, reflection) cannot map one triangle into the other.

The transformations are not a sequence of rigid transformations as the sides of the triangles differ significantly, which indicates possible sizes and shapes that are different. Hence, they could be related by a dilation (scaling), but not by congruent transformations.

This reasoning reflects the application of the Side-Side-Side (SSS) congruence postulate: all sides must be equal for the triangles to be congruent. In this case, since we found unequal lengths, we conclude that triangle ABC cannot be transformed into triangle XYZ through rigid transformations.