In this chapter we will learn method to rotate a given body counterclockwise by 90 degree about the origin.
Let us first review the basics of rotation.
What is Rotation ?
When a body is moved around the central point such that the distance between them is always the same then the movement is called Rotation.
For example;
The movement of earth around the sun is called Rotation.
Note that here the central point is the sum and full 360 degree rotation results in circle.
Moving point in 90 degree counterclockwise rotation
Here we will learn the rotation of point in counterclockwise direction by 90 degree around a fixed point called origin.
Consider a point A located at coordinate (-3, 2).
We will rotate this point by 90 degree counterclockwise direction around origin (0, 0)
Let’s find the angle measurement of point A with horizontal axis.
Hence, the point A forms 33 degree angle with the horizontal axis.
i.e. ∠AOX = 33 degree
Since we need to rotate the point by 90 degrees, we would extend the above angle by another 57 degree.
if we combine both the given angle, we will get angle with 90 degree.
i.e. ∠AOM = 90 degree
Now take a divider and set its length equal to OA.
Place one leg of divider on point O and cut an arc on line OM.
Hence point B (-2, -3) is the final position of point A rotated 90 degree in counterclockwise direction.
I hope you understood the above process. Let us look at shortcut method to identify location of point rotated by 90 degrees counterclockwise direction.
Shortcut for 90 degree counterclockwise rotation
You can easily find the final location of rotated point by using following method.
If point (h, k) is rotated 90 degree counterclockwise, then the final position of point will be (-k, h)
Hence,
If original point ⟹ (h, k) ;
then 90 degree counterclockwise rotated point ⟹ (-k, h)
Let us see some solved examples for further understanding.
Example 01
The point (3, 4) is rotated 90 degree counterclockwise around origin. Find the location of rotated point.
Solution
Original Point Coordinate ⟹ (3, 4)
Rotated point coordinate ⟹ (-4, 3)
Given below is the graphical representation of above rotation.
Example 02
The point (-3, 2) is rotated 90 degree counterclockwise direction. Find the location of rotated point.
Solution
Original Point Coordinate ⟹ (-3, 2)
Rotated point coordinate ⟹ (-2, -3)
Given below is the graphical representation of above rotation.
Example 03
The point (2, -2) is rotated 90 degrees counterclockwise direction. Find the location of final point.
Solution
Original Point Coordinate ⟹ (2, -2)
Rotated point coordinate ⟹ (2, 2)
Showing the rotation in the form of graphical image.
90 degree counterclockwise rotation of object
You can rotate the simple geometrical figures by following the below steps.
(a) Locate the position of all vertices.
(b) Now rotate each of the vertices individually.
(c) Find the coordinates of all rotated points using above mentioned shortcut formula
(d) join all the point to form complete figure.
Let us see below example to understand the process.
Example 01
Rotate the below triangle in 90 degree counterclockwise direction around the origin.
Solution
Since the points are rotated 90 degree counterclockwise, we will use the above mentioned shortcut formula.
Point (-5, 5) will rotate to ⟹ (-5, -5)
Point (-6, 3) will rotate to ⟹ (-3, -6)
Point (-3, 4) will rotate to ⟹ (-4, -3)
Plotting all the points in graphical figure and joining them.
