Read the following

Question

To determine the transformations made to triangle \( \triangle ABC \) to produce triangle \( \triangle XYZ \) and to assess the congruence of the two triangles, we can follow these steps:

### 1. Find the Vertices of Each Triangle:

- **Triangle \( \triangle ABC \)** has vertices:
- \( A(2, -5) \)
- \( B(2, -3) \)
- \( C(5, -3) \)

- **Triangle \( \triangle XYZ \)** has vertices:
- \( X(0, -6) \)
- \( Y(0, -4) \)
- \( Z(-3, -4) \)

### 2. Calculate Distances Between Points:

We will calculate the lengths of the sides of both triangles to compare them to see if they are congruent.

**For \( \triangle ABC \):**
- Length of \( AB \):
\[
AB = \sqrt{(2-2)^2 + (-5 + 3)^2} = \sqrt{0 + 4} = 2
\]
- Length of \( BC \):
\[
BC = \sqrt{(2-5)^2 + (-3 + 3)^2} = \sqrt{(-3)^2 + 0} = 3
\]
- Length of \( CA \):
\[
CA = \sqrt{(2-5)^2 + (-5 + 3)^2} = \sqrt{(-3)^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13}
\]

**For \( \triangle XYZ \):**
- Length of \( XY \):
\[
XY = \sqrt{(0-0)^2 + (-6 + 4)^2} = \sqrt{0 + 4} = 2
\]
- Length of \( YZ \):
\[
YZ = \sqrt{(0 + 3)^2 + (-4 + 4)^2} = \sqrt{3^2 + 0} = 3
\]
- Length of \( ZX \):
\[
ZX = \sqrt{(0 + 3)^2 + (-6 + 4)^2} = \sqrt{3^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13}
\]

### 3. Compare Side Lengths:

Now let's summarize the side lengths:

- For \( \triangle ABC \):
- \( AB = 2 \)
- \( BC = 3 \)
- \( CA = \sqrt{13} \)

- For \( \triangle XYZ \):
- \( XY = 2 \)
- \( YZ = 3 \)
- \( ZX = \sqrt{13} \)

### 4. Assess Congruence:

Since all corresponding side lengths are equal:
- \( AB = XY \)
- \( BC = YZ \)
- \( CA = ZX \)

We can conclude that \( \triangle ABC \) is congruent to \( \triangle XYZ \) by the Side-Side-Side (SSS) congruence criterion.

### 5. Determine Transformations:

To find the transformations, we can observe the coordinates of the vertices:

- **Translation**:
To translate \( A(2,-5) \) to \( X(0,-6) \), we subtract \( (2, -5) \) from \( (0, -6) \):
\[
\text{Translation vector} = (0 - 2, -6 + 5) = (-2, -1)
\]

Applying the same translation to points B and C:
- \( B(2,-3) \) becomes \( (2 - 2, -3 -1) = (0,-4) \) which is the point \( Y \).
- \( C(5,-3) \) becomes \( (5 - 2, -3 - 1) = (3,-4) \), noting that \( 3 \) in the negative direction gives \( Z(-3, -4) \).

Thus, the transformation includes a translation by vector \((-2, -1)\).

### Conclusion:

In conclusion, \( \triangle ABC \) is congruent to \( \triangle XYZ \) through a transformation that consists of a translation by the vector \((-2, -1)\).

Now, a composition transformation to place, so two different types of transformations occured. What type of transformation would turn triangle, and by how many degrees?

Answer 1

To determine the composition of transformations required to transform triangle \( \triangle ABC \) into triangle \( \triangle XYZ \), we will analyze the situation further.

Given Transformations:

From the analysis provided earlier, we concluded that one of the transformations is a translation by the vector \((-2, -1)\). This means that each point in triangle \( \triangle ABC \) is moved down and to the left by these respective amounts.

Identifying Additional Transformations:

To assess whether any rotation or reflection is needed, we will investigate the orientation and relative positioning of the triangles after the translation.

1. Vertices After Translation:

After applying the translation vector \((-2, -1)\) to each vertex of \( \triangle ABC \):

• Translate \( A(2, -5) \) to \( A'(0, -6) \) (which is \( X \)).
• Translate \( B(2, -3) \) to \( B'(0, -4) \) (which is \( Y \)).
• Translate \( C(5, -3) \) to \( C'(3, -4) \) (but we find that this should align with \( Z(-3, -4) \) correctly, thus needs further examination).

Observing the Orientation:

The original triangle \( \triangle ABC \) has points:

• \( A(2, -5) \) at the top
• \( B(2, -3) \) further down (directly below \( A \))
• \( C(5, -3) \) to the right of point \( B \)

After translating this, \( A' \) and \( B' \) remain directly above one another, which aligns with points \( X \) and \( Y \). However, point \( C \) needs further adjustment to align with \( Z(-3, -4) \).

Determining Rotation:

To move \( C'(3, -4) \) to point \( Z(-3, -4) \):

• A reflection across the vertical line \( x = 0 \) willhelp achieve this horizontal displacement.

Thus, after translating \( \triangle ABC \) to align its first two vertices with \( \triangle XYZ \), we could then apply a reflection across the vertical axis \( x = 0 \). This is not merely a rotation, but a reflection that further aligns point \( C \) to \( Z \).

Conclusion:

Therefore, the transformations that are needed to obtain triangle \( \triangle XYZ \) from triangle \( \triangle ABC \) includes:

1. A Translation by vector \((-2, -1)\).
2. A Reflection across the vertical axis \( x = 0 \) that allows point \( C \) to match with point \( Z \).

So, to conclude, the contribution is a combination of a translation and a reflection rather than solely a rotation. No rotation by degrees is involved in this transformation.