Rotations can be applied to points and shapes in the coordinate plane using coordinate notation. To perform a rotation, we need to understand two key components: the origin and the angle of rotation.
The origin is the fixed point around which the rotation occurs. It serves as the center of rotation. The angle of rotation determines the amount and direction of the rotation. A positive angle represents a counterclockwise rotation, while a negative angle represents a clockwise rotation.
To apply a rotation to a point or shape, we follow these steps:
Find the coordinates of the point or vertices of the shape.
Identify the origin of the rotation.
Determine the angle of rotation.
Use the rotation formula to find the new coordinates:
If the point has coordinates (x, y) and the origin has coordinates (h, k) with an angle of rotation θ, the new coordinates (x', y') can be calculated as:
x' = (x - h) * cos(θ) - (y - k) * sin(θ) + h y' = (x - h) * sin(θ) + (y - k) * cos(θ) + k
Let's consider an example. Suppose we want to rotate the point A(3, 2) by an angle of 90 degrees counterclockwise around the origin O(0, 0). Using the rotation formula, we can calculate the new coordinates of A as follows:
x' = (3 - 0) * cos(90°) - (2 - 0) * sin(90°) + 0 = 2 y' = (3 - 0) * sin(90°) + (2 - 0) * cos(90°) + 0 = 3
Therefore, the new coordinates of A after rotation are A'(2, 3).
Remember, practicing with various examples will help solidify your understanding of applying rotations in the coordinate plane. So, grab a pencil and get ready to rotate some points and shapes!
Keep up the great work, and remember, practice makes perfect. You're well on your way to mastering rotations in the coordinate plane!