The point (-5,-4) is rotated 90 counterclockwise .
The correct answer and explanation is:
Correct Answer:
The point (-5, -4) rotated 90 degrees counterclockwise becomes (4, -5).
Explanation:
Rotating a point in the coordinate plane involves changing the location of the point based on a specific rule. For a 90-degree counterclockwise rotation around the origin (0, 0), the rule is:
(x, y) → (-y, x)
This means you take the y-coordinate of the original point, negate it, and make it the new x-coordinate. Then you take the x-coordinate and make it the new y-coordinate.
Given the original point is (-5, -4):
- The original x = -5
- The original y = -4
- Applying the rule:
- The new x-coordinate becomes -(-4) = 4
- The new y-coordinate becomes -5
- So the rotated point is (4, -5)
This transformation is part of rigid motions in geometry, which means the shape or size of a figure does not change, only its orientation or position. Rotations specifically turn a point or shape around a central point, usually the origin, without altering distances or angles.
Understanding this rotation helps in various applications. In algebra and geometry, transformations such as rotations are used to analyze symmetry, design objects in engineering, and develop computer graphics. In more advanced topics, such transformations are essential in linear algebra, where rotation matrices generalize this concept in higher dimensions.
By mastering the basic rotation rules, especially around the origin, one builds a foundation for solving more complex geometric and algebraic problems. The ability to mentally visualize or compute these rotations is also beneficial in spatial reasoning and real-world tasks like navigation or architectural design.