To rotate an object 90 degrees counterclockwise about the origin, you apply a specific transformation rule to every point that makes up the object.
Understanding 90-Degree Counterclockwise Rotation
Rotating an object in a coordinate plane involves rotating each individual point of that object. The most common type of rotation is about the origin (0,0). The rule for rotating a point 90 degrees counterclockwise about the origin is fundamental to understanding object rotation.
According to the provided information, when rotating a point A with coordinates (x, y) 90 degrees counterclockwise about the origin, the new point A' will have coordinates (-y, x).
The Transformation Rule: (x, y) → (-y, x)
This rule can be easily remembered:
- Switch the positions of the original x and y coordinates.
- Make the new first coordinate (which was the original y) negative.
In other words, as stated in the reference, switch x and y and make y negative (referring to the original y coordinate becoming the negative of the new x coordinate).
Applying the Rule to an Object
For an object, such as a polygon defined by its vertices, you rotate the entire object by applying this transformation rule to each of its vertices. The new, rotated vertices will form the rotated object.
Steps to Rotate an Object
- Identify the coordinates (x, y) of each key point or vertex of the object.
- For each point (x, y), apply the 90-degree counterclockwise rotation rule to find its new coordinates (-y, x).
- Connect the new points in the same order as the original points to form the rotated object.
Example: Rotating a Triangle
Let's rotate a triangle with vertices at A(1, 2), B(4, 1), and C(2, 4) 90 degrees counterclockwise about the origin.
- Point A (1, 2):
- Switch x and y: (2, 1)
- Make the new first coordinate (original y) negative: (-2, 1)
- So, A' is at (-2, 1).
- Point B (4, 1):
- Switch x and y: (1, 4)
- Make the new first coordinate (original y) negative: (-1, 4)
- So, B' is at (-1, 4).
- Point C (2, 4):
- Switch x and y: (4, 2)
- Make the new first coordinate (original y) negative: (-4, 2)
- So, C' is at (-4, 2).
The rotated triangle will have vertices at A'(-2, 1), B'(-1, 4), and C'(-4, 2).
By applying the simple rule of transforming each point from (x, y) to (-y, x), you effectively rotate the entire object 90 degrees counterclockwise around the origin.
