Rotation Formula
The rotation formula tells us about the rotation of a point with respect to the origin. The rotation formula is used to find the position of the point after rotation. Rotation is a circular motion around the particular axis of rotation or point of rotation. In general, rotation can be done in two common directions, clockwise and anti-clockwise or counter-clockwise direction. Let us learn the rotation formula along with a few solved examples.
What Is Rotation Formula?
The rotation formula will give us the exact location of a point after a particular rotation to a finite degree of rotation. The rotation formula depends on the type of rotation done to the point with respect to the origin. There are four major types of transformation that can be done to a geometric two-dimensional shape.
The rotation formula according to the type of rotation done is shown in the table given below:
| Rotation | Coordinates of the point | Coordinates of the Image of the point after Rotation |
|---|---|---|
| Rotation of 90°(CW) | (x, y) | (y, -x) |
| Rotation of 90°(CCW) | (x, y) | (-y, x) |
| Rotation of 180° | (x, y) | (-x, -y) |
| Rotation of 270°(CW) | (x, y) | (-y, x) |
| Rotation of 270°(CCW) | (x, y) | (y, -x) |
Let us see the applications of the rotation formula in the following solved examples.
Solved Examples Using Rotation Formula
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Example 1: Find the position of the point K(5, 7) after the rotation of 90°(CCW) using the rotation formula.
Solution:
Using the rotation formula,
After rotation of 90°(CCW), coordinates of the point (x, y) becomes: (-y, x)
Hence the point K(5, 7) will have the new position at (-7, 5)Answer: Therefore, the coordinates of the image are (-7, 5).
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Example 2: Find the position of point A, if the coordinates of the images of the point after the rotation of 270°(CW) is (-9, -7), using the rotation formula.
Solution:
Using the rotation formula,
After rotation of 270°(CW), coordinates of the point (x, y) becomes: (-y, x)
Hence the point A(x, y) will have the new position at (-9, -7) if the point was initially at (7, -9)Answer: Therefore, the coordinates of point A are (7, -9).
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