# Rotation 180 degrees about the origin

In this chapter we will learn how to rotate the body by 180 degrees in clockwise and anticlockwise direction about the origin.

We will also look at the solved examples for better understanding of the concept.

## Rotating point by 180 degree about origin

Let us first rotate the point by 180 degrees.

Whether the point is rotated clockwise or counter-clockwise, the final position of point after 180 degree rotation will be the same.

Consider the above point A (3, 4).

Let’s rotate the point by 180 degree clockwise direction.

First find the angle for point A with respect to horizontal axis.

Here the point A forms 53 degree angle with horizontal axis.

i.e. ∠AOX = 53 degree

In order to make 180 degree rotation, we have to extend 127 degree more.

i.e. 180 – 53 = 127 degree.

Here ∠XOY = 127 degree.

If we join both the angles we will get 180 degrees.

Now take a divider and set it to length OA.

Place one leg of divider at origin O and cut an arc on line y.

Hence, the point A (3, 4) is rotated by 180 degree to reach point B (-3, -4).

Let us now see the shortcut method to directly located the final position of point rotated by 180 degree.

### Shortcut for 180 degree clockwise / counterclockwise rotation

If a point is rotated by 180 degree in clockwise / counterclockwise rotation, the location of final point is given by following method.

If (h, k) is the initial point, then after 180 degree rotation the location of final point will be (-h, -k).

Note that in 180 degree rotation, both clockwise & anticlockwise rotation results in same final point.

Hence,
Original point ⟹ (h, k)
180 degree rotated point ⟹ (-h, -k)

Let us see some solved examples for better conceptual understanding.

Example 01
The point (3, -5) is rotated 180 degree clockwise. Find the coordinates of rotated point.

Solution
Initial Point ⟹ (3, -5)
Final Rotated Point ⟹ (-3, 5)

Plotting the rotation in graphical image.

Example 02
The point (5, 2) is rotated 180 degree counterclockwise around origin. Find the coordinates of rotated point.

Solution
Initial Point ⟹ (5, 2)
Final Rotated Point ⟹ (-5, -2)

Plotting the rotation in graphical image.

Example 03
The point (-3, -3) is rotated by 180 degree clockwise rotation around origin. Find the location of final point.

Solution
Initial Point ⟹ (-3, -3)
Final Rotated Point ⟹ (3, 3)

Plotting the rotation in graphical diagram.

### How to rotate the object by 180 degree ?

You can rotate the simple graphical figure in cartesian coordinate by following below steps;

(a) Locate the position of all vertices

(b) Now rotate each vertices by 180 degree individually.

(c) Locate the position of rotated vertices by using above shortcut method.

(d) Join all the rotated points to form the figure.

Given below is solved example for further understanding.

Example 01
Rotate the below quadrilateral by 180 degree clockwise direction.

Solution
Since the body is rotated by 180 degree around origin, we will apply the above mentioned shortcut method for individual vertices.

Point (-5, 3) rotated to ⟹ (5, -3)
Point (-3, 4) rotated to ⟹ (3, -4)
Point (-2, 2) rotated to ⟹ (2, -2)
Point (-4, 1) rotated to ⟹ (4, -1)

Plotting the rotation in graphical diagram.