Formula For 180 Degree Rotation

Before learning the formula for 180-degree rotation, let us recall what is 180 degrees rotation. A point in the coordinate geometry can be rotated through 180 degrees about the origin, by making an arc of radius equal to the distance between the coordinates of the given point and the origin, subtending an angle of 180 degrees at the origin. We have to rotate the point about the origin with respect to its position in the cartesian plane. It can be well understood in the following section of the formula for 180-degree rotation.

What is the Formula for 180 Degree Rotation?

The formula for 180-degree rotation of a given value can be expressed as if R(x, y) is a point that needs to be rotated about the origin, then coordinates of this point after the rotation will be just of the opposite signs of the original coordinates. i.e., the coordinates of the point after 180-degree rotation are:

R'= (-x, -y) 

Let us apply the formula for 180-degree rotation in the following solved examples.

Examples Using Formula for 180 Degree Rotation

Example 1: Rotate the following points by 180 degrees: (i) A(3,4) (ii) B(2,-7) (iii) C(-5, -1).

Solution:

To find: Rotate the given points by 180 degrees.
Given: A(3,4), B(2.-7), C(-5,-1)

Using formula for 180 degree rotation,
R(x,y) ⇒ R'(-x,-y)

(i). A(3,4)  ⇒ A’(-3,-4)
(ii). B(2,-7) ⇒ B’(-2,7)
(iii).C(-5,-1) ⇒ C’(5,1)

Answer:  A’(-3,-4), B’(-2,7), and C’(5,1) are the 180 degrees rotated points of A(3,4), B(2.-7), and C(-5,-1)

Example 2: Rotate a line AB having ends as A(4,5) and B(-1,2) by 180 degrees.

Solution:

To find: Rotate a line AB by 180 degrees.
Given
Line AB, A(4,5), B(-1,2)

Using formula for 180 degree rotation,

R(x,y) ⇒ R'(-x,-y)

For A(4,5) ⇒ A’(-4,-5)
For  B(-1,2) ⇒ B’(1,-2)

Answer: The coordinates of the new line A’B’ is A’(-4,-5) and B’(1,-2)