Question

A triangle with points 1, -1 3, -2 and 1 negative 3 rotated 90 degrees counter clockwise What is the translation To -4, 2 -3, 4 and -2, 2

Answer 1

To apply a 90-degree counterclockwise rotation and then perform a translation, we will follow these steps:

1. **Identify the vertices of the triangle**. The vertices are:
- \( A(1, -1) \)
- \( B(3, -2) \)
- \( C(1, -3) \)

2. **Perform the 90-degree counterclockwise rotation**. The counterclockwise rotation transformation can be expressed mathematically as:
\[
(x, y) \rightarrow (-y, x)
\]

Applying this transformation to each vertex:
- For \( A(1, -1) \):
\[
A'(1, -1) \to (1, -1) \rightarrow (1, -1) \to (-(-1), 1) = (1, 1)
\]

- For \( B(3, -2) \):
\[
B(3, -2) \to (-(-2), 3) = (2, 3)
\]

- For \( C(1, -3) \):
\[
C(1, -3) \to (-(-3), 1) = (3, 1)
\]

Now, our rotated vertices are:
- \( A'(1, 1) \)
- \( B'(2, 3) \)
- \( C'(3, 1) \)

3. **Perform the translation**. The translation points we want to translate to are:
- \( D(-4, 2) \)
- \( E(-3, 4) \)
- \( F(-2, 2) \)

We must establish a translation vector that will move the triangle's vertices to these new positions.

We can apply translations to each vertex to move them. Let's consider translating the points such that \( A'(1, 1) \) moves to \( D(-4, 2) \).

To get the translation vector:
\[
\text{Translation vector} = D - A' = (-4, 2) - (1, 1) = (-4 - 1, 2 - 1) = (-5, 1)
\]

We will apply this translation vector \((-5, 1)\) to each of the rotated vertices:
- For \( A'(1, 1) \):
\[
A''(1, 1) + (-5, 1) = (1 - 5, 1 + 1) = (-4, 2)
\]

- For \( B'(2, 3) \):
\[
B''(2, 3) + (-5, 1) = (2 - 5, 3 + 1) = (-3, 4)
\]

- For \( C'(3, 1) \):
\[
C''(3, 1) + (-5, 1) = (3 - 5, 1 + 1) = (-2, 2)
\]

After completing these calculations, the translation effectively moves the triangle to the new points:
- New vertex \( A''(-4, 2) \)
- New vertex \( B''(-3, 4) \)
- New vertex \( C''(-2, 2) \)

### Final Result:
The original triangle is rotated 90 degrees counterclockwise, followed by a translation to the points:

- \((-4, 2)\)
- \((-3, 4)\)
- \((-2, 2)\)