https://file.aiquickdraw.com/mathbot/file/17272978049171wpwfzep.jpg

The problem involves rotating triangle \( ABC \) counterclockwise by 270° about the origin to obtain triangle \( DEF \). The coordinates of the vertices of triangle \( ABC \) are given as:

- \( A(1, 3) \)
- \( B(1, 1) \)
- \( C(4, 1) \)

To rotate a point by 270° counterclockwise about the origin, the transformation rule is as follows:

\[
(x, y) \rightarrow (y, -x)
\]

Let's apply this rule to each vertex of triangle \( ABC \).

1. **For point \( A(1, 3) \):**
   \[
   A(1, 3) \rightarrow (3, -1)
   \]
   So, \( A \rightarrow D(3, -1) \).

2. **For point \( B(1, 1) \):**
   \[
   B(1, 1) \rightarrow (1, -1)
   \]
   So, \( B \rightarrow E(1, -1) \).

3. **For point \( C(4, 1) \):**
   \[
   C(4, 1) \rightarrow (1, -4)
   \]
   So, \( C \rightarrow F(1, -4) \).

Thus, the coordinates of triangle \( DEF \), which is the rotated image of triangle \( ABC \), are:

- \( D(3, -1) \)
- \( E(1, -1) \)
- \( F(1, -4) \)

Would you like further details on rotation transformations or the visualization of this transformation?

Here are five related questions:
1. How does the rotation transformation change for different angles?
2. What are the coordinates if the triangle is rotated by 90° instead of 270°?
3. How can you verify the accuracy of a rotation transformation using a graph?
4. What happens to the area of a triangle after a rotation?
5. Can a rotation transformation result in a different type of triangle (e.g., equilateral to isosceles)?

**Tip:** Always double-check transformations by graphing the preimage and image to ensure accuracy.

Preimage: A(1, 3), B(1, 1), C(4, 1), image: triangle DEF, rotation of ABC 270° counterclockwise about the origin.

Learn how to rotate triangle ABC with vertices A(1, 3), B(1, 1), and C(4, 1) counterclockwise by 270° about the origin to get the image triangle DEF. Apply the coordinate transformation rule (x, y) → (y, -x) to find the new coordinates of D, E, and F. Suitable for students in Grades 8-10.

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