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Identify the coordinates of the vertices of the figure *ABCD* after the rotation of 270o about the origin and then 90o rotation about the origin.

To find the coordinates of the vertices of the figure \(ABCD\) after performing a \(270^\circ\) rotation about the origin followed by a \(90^\circ\) rotation about the origin, we can proceed with the following steps:

### Step 1: Identify the Original Coordinates

Let's first identify the coordinates of the vertices \(A\), \(B\), \(C\), and \(D\) from the image:

- \( A \) is approximately at \((-3, 3)\).
- \( B \) is approximately at \((3, 4)\).
- \( C \) is approximately at \((2, -2)\).
- \( D \) is approximately at \((-2, -3)\).

### Step 2: Apply the \(270^\circ\) Rotation About the Origin

A \(270^\circ\) rotation about the origin is equivalent to a \(90^\circ\) clockwise rotation, and the transformation is as follows:

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

Applying this to each vertex:

- \( A(-3, 3) \rightarrow (3, 3) \)
- \( B(3, 4) \rightarrow (4, -3) \)
- \( C(2, -2) \rightarrow (-2, -2) \)
- \( D(-2, -3) \rightarrow (-3, 2) \)

### Step 3: Apply the \(90^\circ\) Rotation About the Origin

Now, we apply the \(90^\circ\) rotation to the coordinates obtained after the \(270^\circ\) rotation. A \(90^\circ\) rotation about the origin is equivalent to the transformation:

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

Applying this to the new coordinates:

- \( A(3, 3) \rightarrow (-3, 3) \)
- \( B(4, -3) \rightarrow (3, 4) \)
- \( C(-2, -2) \rightarrow (2, -2) \)
- \( D(-3, 2) \rightarrow (-2, -3) \)

### Final Coordinates After Both Rotations

The final coordinates of the vertices \(A\), \(B\), \(C\), and \(D\) after both rotations are:

- \( A \rightarrow (-3, 3) \)
- \( B \rightarrow (3, 4) \)
- \( C \rightarrow (2, -2) \)
- \( D \rightarrow (-2, -3) \)

Interestingly, the final coordinates match the original coordinates, which indicates that the composition of a \(270^\circ\) and a \(90^\circ\) rotation about the origin results in the identity transformation.

Would you like more details or have any questions? Here are some related questions to explore further:

1. What would happen if you rotated the figure \(ABCD\) by \(180^\circ\) instead of \(270^\circ\)?
2. How do the coordinates of the vertices change after a \(360^\circ\) rotation?
3. What is the effect of reflecting the figure over the x-axis and then performing a rotation?
4. How can you determine the center of rotation given the original and rotated figures?
5. How do you calculate the area of the quadrilateral \(ABCD\) after rotation?

**Tip:** When performing consecutive transformations, always apply each transformation step by step, ensuring that each result is used accurately in the next step.

Explore how to find the coordinates of vertices after rotating the figure ABCD by 270° and then 90° about the origin. Step-by-step solution with original and final coordinates provided.

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