The problem asks about rotating a line segment $AB$ through a certain angle around the point $R(0, -2)$. The key steps involve finding the coordinates of points after rotation, the equation of the line, and understanding the different angles of rotation.

Here’s the breakdown:

1. Identify Points $A$ and $B$:

   • Point $A = (1, -4)$
   • Point $B = (3, 4)$

2. Center of Rotation $R$:

   • Point $R = (0, -2)$

3. Angles of Rotation:

   • The rotation angles mentioned are $\theta = -90^\circ$, $90^\circ$, and $180^\circ$.

4. Steps for Rotation:

   • To rotate points $A$ and $B$ around $R$, use rotation formulas:
     • For rotation by an angle $\theta$ around a point $(x_0, y_0)$, the new coordinates $(x', y')$ of a point $(x, y)$ can be calculated as: $x' = x_0 + (x - x_0) \cos \theta - (y - y_0) \sin \theta$ $y' = y_0 + (x - x_0) \sin \theta + (y - y_0) \cos \theta$

5. Equation of the Line $AB$:

   • Calculate the slope and intercept of the line $AB$ if required.

Would you like detailed calculations for each rotation angle or the equation of the line through points $A$ and $B$?

Related Questions:

1. How do you calculate the coordinates of a point after rotation?
2. What is the equation of a line passing through two points?
3. How does the center of rotation affect the rotated coordinates?
4. What is the impact of a $180^\circ$ rotation on a line segment?
5. How can you confirm if a rotation has been applied correctly?

Tip:

For quick rotations around the origin by angles like $90^\circ$ or $180^\circ$, try using simple coordinate rules (e.g., $90^\circ$ rotation switches $(x, y)$ to $(-y, x)$).