https://file.aiquickdraw.com/mathbot/file/172666557130859x0j0gr.jpg

To analyze the transformation that Marc applied to triangle \( \triangle ABC \) to get \( \triangle A'B'C' \), let's first carefully examine both triangles based on the coordinate grid in the image.

1. **Vertices of the Original Triangle \( \triangle ABC \)**:
   - \( A \) is located at the origin \( (0,0) \).
   - \( B \) appears to be at \( (-4, -3) \).
   - \( C \) appears to be at \( (-2, -4) \).

2. **Vertices of the Transformed Triangle \( \triangle A'B'C' \)**:
   - \( A' \) is located at \( (4,5) \).
   - \( B' \) is located at \( (2,2) \).
   - \( C' \) is located at \( (3,0) \).

### Sequence of Transformations:

#### Step 1: **Translation**
To get from the origin \( A(0,0) \) to \( A'(4,5) \), Marc likely applied a **translation** of \( 4 \) units to the right and \( 5 \) units up.

#### Step 2: **Rotation**
Next, observe that \( \triangle A'B'C' \) is rotated compared to \( \triangle ABC \). The orientation suggests a counterclockwise rotation of **90 degrees** about point \( A' \). After the translation, this rotation aligns \( \triangle ABC \) to the final position of \( \triangle A'B'C' \).

### Conclusion:
The sequence of transformations Marc used is:
1. A translation of \( 4 \) units to the right and \( 5 \) units up.
2. A counterclockwise rotation of \( 90^\circ \) about the point \( A' \).

Would you like further details or clarifications on the transformation? Here are some additional questions for deeper understanding:

1. How can we determine the rotation angle between two triangles?
2. What are other possible transformations that could result in a similar triangle?
3. How do we verify that the triangles are indeed similar after transformations?
4. Could the transformation include a reflection instead of a rotation? Why or why not?
5. What changes in the orientation of triangle \( \triangle ABC \) suggest a counterclockwise rotation?

**Tip**: When identifying transformations, always track the movement of at least two points to confirm the type of transformation (translation, rotation, reflection, or dilation).

Marc graphed ΔABC on the coordinate grid as shown, then applied a series of transformations that produced a similar triangle labeled A'B'C'. What sequence of transformations did Marc use to produce triangle A'B'C'?

Explore the geometric transformations applied by Marc to triangle ABC, including translation and rotation, to produce a similar triangle A'B'C'. This problem involves understanding coordinate geometry and the properties of similar triangles. Suitable for students in Grades 8-10.

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