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The problem asks to find the image of the point \((-5, 3)\) after a **270° rotation** about the point \((-7, -3)\).

To solve this, we'll follow these steps:

### Step 1: Translate the points.
We first translate the point of rotation \((-7, -3)\) to the origin \((0, 0)\) by shifting the entire system.

To translate a point \((x, y)\) by a vector \((a, b)\), we subtract \((a, b)\) from \((x, y)\):

- Translation of \((-5, 3)\) with respect to \((-7, -3)\):
  \[
  (-5, 3) - (-7, -3) = (-5 + 7, 3 + 3) = (2, 6)
  \]

Now, we have the translated point \((2, 6)\).

### Step 2: Apply the 270° rotation.
A 270° counterclockwise rotation is equivalent to a 90° clockwise rotation. The general formula to rotate a point \((x', y')\) by 270° counterclockwise (or 90° clockwise) around the origin is:
\[
(x', y') \rightarrow (y', -x')
\]

Thus, applying this formula to the point \((2, 6)\):
\[
(2, 6) \rightarrow (6, -2)
\]

### Step 3: Translate back to the original position.
Now, we reverse the initial translation by adding \((-7, -3)\) back to the rotated point \((6, -2)\):
\[
(6, -2) + (-7, -3) = (6 - 7, -2 - 3) = (-1, -5)
\]

### Final Answer:
The image of the point \((-5, 3)\) after a 270° rotation about the point \((-7, -3)\) is **\((-1, -5)\)**.

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Do you need more details or have any other questions?

Here are 5 related questions:
1. How would you perform a 90° rotation around a given point?
2. What is the general formula for rotating a point by any angle?
3. How does the sign of the angle (clockwise vs. counterclockwise) affect the rotation?
4. What happens if the center of rotation is the origin instead of a different point?
5. How can you use matrices to perform rotations in two dimensions?

**Tip:** When performing rotations around a point that isn't the origin, always translate the point of rotation to the origin, apply the rotation, and then translate back to the original position.

What is the image of the point (-5, 3) under a 270° rotation about the point (-7, -3)?

Learn how to calculate the image of the point (-5, 3) under a 270° rotation around the point (-7, -3). This detailed explanation covers coordinate geometry, rotation formulas, and transformations. Suitable for high school students in Grades 9-12.

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