Solved Examples on Rotational Formula
Example 1: Calculate the coordinates of the point (5, 3) after rotating 90° clockwise.
Solution:
Given, (x, y) = (5, 3)
After rotating the point 90° clockwise the coordinates are given using (y, -x)
Thus, the coordinates of the point after rotation are (3, -5).
Example 2: Calculate the coordinates of the point (3, 4) after rotating 180° anti-clockwise.
Solution:
Given, (x, y) = (3, 4)
After rotating the point 180° anti-clockwise the coordinates are given using (-x, -y)
Thus, the coordinates of the point after rotation are (-3, -4).
Example 3: The coordinates of the point (x, y) after rotating 270° clockwise are (3, 8). What are the actual coordinates of the point?
Solution:
Given, initial coordinates as (x, y)
Final coordinates after rotation are (3, 8). We know that final coordinates after rotating 270° clockwise are given by (-y, x).
Thus (-y, x) = (3, 8)
So (x, y) = (8, -3)
Thus, the actual coordinates of the point are (8, -3).
Example 4: Find the coordinates of the point (1, -6) after rotating 90° anti-clockwise.
Solution:
Given, (x, y) = (1, -6)
After rotating the point 90° clockwise the coordinates are given using (-y, x)
Thus, the coordinates of the point after rotation are (6, 1).
Example 5: Find the coordinates of point (4, 5) when it is rotated 45° anti-clockwise around point Q(5, 5).
Solution:
Given, (x, y) = (4, 5), θ = 45°(positive because the rotation is anti-clockwise), (α, β) = (5, 5)
Using the rotation formula when the point is rotated about a given point:
(x’, y’) = (α + (x – α) cosθ – (y – β) sinθ, β + (x – α) sinθ – (y – β) cosθ)
= (5 + (4 – 5) cos45° – (5 – 5) sin45°, 5 + (4 – 5) sin45° – (5 – 5) cos45°)
= (5 – 1/√2 – 0, 5 – 1/√2 – 0)
Thus, (x’, y’) = (5 – 1/√2, 5 – 1/√2)
Rotation
In real life, we know that the Earth rotates on its own axis and the moon also rotates on its axis. But what basically rotation is? Also, geometry deals with four basic types of transformations that are Rotation, Reflection, Translation, and Resizing. In this article, we shall read about the fundamental concept of rotation.
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