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180 Degree Rotation

Introduction

Everywhere you turn, there are rotations. The most prevalent example is the earth, which revolves around an axis. An example of a transformation is a rotation, which revolves a figure around a point. The shape and dimensions of a figure remain the same while facing in a different direction. You can rotate a figure either clockwise or counterclockwise.

One of the simplest and most common transformations in geometry is the 180-degree rotation, both clockwise and counterclockwise. 

We will learn more about the 180-degree rotation of a point and a closed figure in this article. The graph before and after the rotation will also be displayed.

What is 180 Degree Rotation?

Definition

A 180-degree rotation transforms a point or figure so that they are horizontally flipped. 

When rotated with respect to the origin, which acts as the reference point, the angle formed between the before and after rotation is 180 degrees. 

A point can be rotated by 180 degrees, either clockwise or counterclockwise, with respect to the origin (0, 0). When this occurs, the new position of point P ( x, y ), denoted by the symbol P’, is (-x, -y). 

If a closed figure is rotated through 180 degrees, the vertices of the original figure will then be considered to identify the new position of the vertices after rotation. A graph is used to illustrate the transformation visually.

The Formula for 180 Degree Rotation

One of the simplest and most common transformations in geometry is the 180-degree rotation, both clockwise and counterclockwise. 

If P (x, y) is a point that must be rotated 180 degrees about the origin, the coordinates of this point after the rotation will only be of the opposite signs of the original coordinates. Below is how the formula for the 180-degree rotation of a given point is represented.

Before Rotation: P ( x , y )
After Rotation: P’ ( -x, -y )

For example, the table below shows the original position of points on a coordinate system and the rotated position through 180 degrees.

Before RotationAfter Rotation
A ( 1, 7 )A ( -1, -7 )
B ( -2, 8 )B ( 2, -8 )
C ( -9, -11 )C ( 9, 11 )
D ( 6, -12 )D ( -6, 12 )
E ( 5, 0 )E ( -5, 0 )
F (0, -3 )F (0, 3 )

Rotating a Point by 180°

A point can be rotated by 180 degrees with respect to the origin, either clockwise or counterclockwise (0, 0). When this happens, the symbol P’ ( -x, -y ) designates the new location of point P (x, y).

Let us have the following points and identify their new position when rotated by 180°.

C ( 2, 4 )
D ( -3, -7 )
E ( -2, 9 )
G ( 5, -1 )

Let us look at the position of each point on a coordinate system. To find the new position of each point after rotation, let us follow the formula ( x, y ) ( -x, -y ).

The new position of point C ( 2, 4 ), when rotated by 180 degrees clockwise or counterclockwise, is C’ ( -2, -4 ).The new position of point D ( -3, -7 ), when rotated by 180 degrees clockwise or counterclockwise, is D’ ( 3, 7 ).
The new position of point E ( -2, 9 ), when rotated by 180 degrees clockwise or counterclockwise, is E’ ( 2, -9 ).The new position of point G ( 5, -1 ) when rotated by 180 degrees clockwise or counterclockwise is G’ ( -5, 1 ).

Rotating a Closed Figure by 180°

The vertices of the original figure will be considered to determine the new position of the vertices after rotation if a closed figure is rotated through 180 degrees. A graph is used to illustrate the transformation visually.

Let us say, for example, points S ( 4, 9 ), T ( 8, 9 ), U ( 4, 1 ), and V ( 8, 1 ) are the vertices of a closed figure which is a rectangle. The image below shows that all points are in Quadrant I.

If we rotate the rectangle by 180° with respect to the origin, we must move its vertices based on the formula ( x, y ) ( -x, -y ).

The new position of vertex S ( 4, 9 ) when rotated by 180 degrees clockwise or counterclockwise is S’ ( -4, -9 ).

The new position of vertex T ( 8, 9 ), when rotated by 180 degrees clockwise or counterclockwise, is T’ ( -8, -9 ).

The new position of vertex U ( 4, 1 ) when rotated by 180 degrees clockwise or counterclockwise is U’ ( -4, -1 ).

The new position of vertex V ( 8, 1 ), when rotated by 180 degrees clockwise or counterclockwise, is V’ ( -8, -1 ).

The graph below shows the given closed figure before and after rotation.

More Examples

Example 1

Find the coordinates of the points that were produced after the points listed below were rotated 180 degrees around the origin.

( a ) L ( 2, 5 )
( b ) M ( -3, 4 )
( c ) N ( -4, -6 )
( d ) S ( 1, -3 )
( e ) P ( -7, 8 )
( f ) Q ( 6, 4 )

Solution

( a ) The new position of point L ( 2, 5 ), when rotated by 180 degrees clockwise or counterclockwise, is L’ ( -2, -5 ). The graph below shows the plotting of point L ( 2, 5 ) and its new position.

( b ) The new position of point M ( -3, 4 ) when rotated by 180 degrees clockwise or counterclockwise is M’ ( 3, -4 ). The graph below shows the plotting of point M ( -3, 4 ) and its new position.

( c ) The new position of point N ( -4, -6 ) when rotated by 180 degrees clockwise or counterclockwise is N’ ( 4, 6 ). The graph below shows the plotting of point N ( -4, -6 ) and its new position.

( d ) The new position of point S ( 1, -3 ) when rotated by 180 degrees clockwise or counterclockwise is S’ ( -1, 3 ). The graph below shows the plotting of point S ( 1, -3 ) and its new position.

( e ) P ( -7, 8 )

The new position of point P ( -7, 8 ) when rotated by 180 degrees clockwise or counterclockwise is P’ ( 7, -8 ). The graph below shows the plotting of point P ( -7, 8 ) and its new position.

( f ) Q ( 6, 4 )

The new position of point Q ( 6, 4 ), when rotated by 180 degrees clockwise or counterclockwise, is Q’ ( -6, -4 ). The graph below shows the plotting of point Q ( 6, 4 ) and its new position.

Example 2

On the graph paper, place the following points and clockwise rotate them 180 degrees about the origin. Determine the new position of each point.

( a ) J (5, -4)
( b ) F ( -3, 10 )
( c ) K ( -2, -6 )
( d ) N ( -9, 7 )

Solution

( a ) J (5, -4)

When point J ( 5, -4 ) is rotated 180 degrees about the origin in the clockwise direction, its new position is J’ ( -5, 4 ). The graph below illustrates that J is in Quadrant IV while J’ ( -5, 4 ) is in Quadrant II.

( b ) F ( -3, 10 )

When point F ( -3, 10 ) is rotated 180 degrees about the origin in the clockwise direction, its new position is F’ ( 3, -10 ). The graph below illustrates that F is in Quadrant II while F’ is in Quadrant IV.

( c ) K ( -2, -6 )

When point K ( -2, -6 ) is rotated 180 degrees about the origin in the clockwise direction, its new position is K’ ( 2, 6 ). The graph below illustrates that K is in Quadrant III while K’ is in Quadrant I.

( d ) N ( -9, 7 )

When point N ( -9, 7 ) is rotated 180 degrees about the origin in the clockwise direction, its new position is N’ ( 9, -7 ). The graph below illustrates that N is in Quadrant II while N’ is in Quadrant IV.

Example 3

On the graph paper, place the following points and counterclockwise rotate them 180 degrees about the origin. Determine the new position of each point.

( a ) T (8, 7)
( b ) R ( -1, 9 )
( c ) I ( -3, -5 )
( d ) Z ( 5, -6 )

Solution

( a ) T (8, 7)

When point T ( 8, 7 ) is rotated 180 degrees about the origin in the counterclockwise direction, its new position is T’ ( -8, -7 ). The graph below illustrates that T is in Quadrant I while T’ ( -8, -7 ) is in Quadrant III.

( b ) R ( -1, 9 )

When point R ( -1, 9 ) is rotated 180 degrees about the origin in the counterclockwise direction, its new position is R’ ( 1, -9 ). The graph below illustrates that RJ is in Quadrant II while R’ ( 1, -9 ) is in Quadrant IV.

( c ) I ( -3, -5 )

When point I ( -3, -5 ) is rotated 180 degrees about the origin in the counterclockwise direction, its new position is I’ ( 3, 5 ). The graph below illustrates that I is in Quadrant III while I’ ( 3, 5 ) is in Quadrant I.

( d ) Z ( 5, -6 )

When point Z ( 5, -6 ) is rotated 180 degrees about the origin in the counterclockwise direction, its new position is Z’ ( -5, 6 ). The graph below illustrates that Z is in Quadrant IV while Z’ ( -5, 6 ) is in Quadrant II.

Example 4

A line segment has the endpoints L ( 2, 5 ) and S ( -6, 8 ). Graph the line segment and rotate it by 180 degrees about the origin.

Solution

When we graph the line segment on a cartesian plane, endpoint L ( 2, 5 ) is in Quadrant I while endpoint S ( -6, 8 ) is in Quadrant II. 

The new position of point L ( 2, 5 ), when rotated by 180 degrees clockwise or counterclockwise, is L’ ( -2, -5 ), while the new position of point S ( -6, 8 ) is S’ ( 6, -8 ).

The graph below shows the line segment LS and its position when rotated by 180 degrees.

Thus, the new position of the line segment of LS is  L’S’.

Example 5

Triangle JKL has the following vertices, J ( 6, -2 ), K ( 1, -8 ), and L ( -6, -5 ). Show the Triangle JKL on the graphing paper and rotate it about the origin through 180°.

Solution

When we plot Triangle JKL, vertex J ( 6, -2 ) is in Quadrant IV, vertex K ( 1, -8 ) is in Quadrant IV, and L ( -6, -5 ) is in Quadrant III. 

If we rotate the triangle by 180° with respect to the origin, we must move its vertices based on the formula ( x, y ) ( -x, -y ).

The new position of vertex J ( 6, -2 ), when rotated by 180 degrees clockwise or counterclockwise, is J’ ( -6, -2 ).

The new position of vertex K ( 1, -8 ), when rotated by 180 degrees clockwise or counterclockwise, is K’ ( -1, 8 ).

The new position of vertex L ( -6, -5 ), when rotated by 180 degrees clockwise or counterclockwise, is L’ ( 6, 5 ).

The graph below shows the before rotation and after rotation of the given closed figure.

Thus, the new position of Triangle JKL is Triangle J’K’L’.

Example 6

A closed figure has the following vertices, A ( 4, 8 ), B ( 10, 2 ), C ( 10, -4 ), and D ( 4, -7 ). Rotate the given closed figure about the origin through 180°.

Solution

If we rotate the closed figure by 180° with respect to the origin, we must move its vertices based on the formula ( x, y ) ( -x, -y ).

The new position of vertex A ( 4, 8 ), when rotated by 180 degrees clockwise or counterclockwise, is A’ ( -4, -8 ).

The new position of vertex B ( 10, 2 ), when rotated by 180 degrees clockwise or counterclockwise, is B’ ( -10, -2 ).

The new position of vertex C ( 10, -4 ) when rotated by 180 degrees clockwise or counterclockwise is C’ ( -10, 4 ).

The new position of vertex D ( 4, -7 ), when rotated by 180 degrees clockwise or counterclockwise, is D’ ( -4, 7 ).

The graph below shows the before rotation and after rotation of the given closed figure.

Thus, the new position of closed figure ABCD is A’B’C’D’.

Summary

One of the simplest and most common transformations in geometry is the 180-degree rotation, both clockwise and counterclockwise. 

A point can be rotated by 180 degrees, either clockwise or counterclockwise, with respect to the origin (0, 0). When this occurs, the new position of point P ( x, y ), denoted by the symbol P’, is (-x, -y).

When rotated with respect to the origin, which acts as the reference point, the angle formed between the before and after rotation is 180 degrees. 

Rotating a Point by 180°

If P (x, y) is a point that must be rotated 180 degrees about the origin, the coordinates of this point after the rotation will only be of the opposite signs of the original coordinates. Below is how the formula for the 180-degree rotation of a given point is represented.

Before Rotation: P ( x , y )
After Rotation: P’ ( -x, -y )

Rotating a Closed Figure by 180°

The vertices of the original figure will be considered to determine the new position of the vertices after rotation if a closed figure is rotated through 180 degrees. A graph is used to illustrate the transformation visually.

Frequently Asked Questions on 180 Degree Rotation ( FAQs )

How do you rotate a closed figure on a graph 180 degrees, either clockwise or counterclockwise?

When a point P ( x, y ) is rotated at 180 degrees either clockwise or counterclockwise about the origin, it assumes a new position P’ ( -x, -y ). Hence, if a closed figure is rotated 180 degrees about the origin, we must get the vertices of the rotated figure and sketch the graph when needed.

What is the 180-degree rotation formula?

If P (x, y) is a point that must be rotated 180 degrees about the origin, the coordinates of this point after the rotation will only be of the opposite signs of the original coordinates. Below is the formula for the 180-degree rotation of a given point.

Before Rotation: P ( x , y )
After Rotation: P’ ( -x, -y )

What is rotation in geometry?

A transformation where a figure is moved around its reference point is called a rotation.

What is meant by 180-degree  rotation?

A 180-degree rotation transforms a point or figure so that they are horizontally flipped. 

When rotated with respect to the origin, which acts as the reference point, the angle formed between the before and after rotation is 180 degrees. 

What is the rule for a 180° clockwise or counterclockwise rotation?

The coordinates of P (x, y) after the rotation will only have the opposite signs of the given coordinates if P needs to be rotated 180 degrees about the origin. Below is the formula for rotating a given point 180 degrees.

Before Rotation: P ( x , y )
After Rotation: P’ ( -x, -y )

What is the difference between clockwise and counterclockwise rotation?

The direction of a turn can be indicated by using either the clockwise or anticlockwise rotation. Anticlockwise involves turning to the left in opposition to the direction of a clock’s hands, while clockwise involves turning to the right.

What is an example of rotating a point by 180°?

The coordinates of P (x, y) after the rotation will only have the opposite signs of the initial coordinates  P needs to be rotated 180 degrees about the origin. This is how the formula for rotating a given point by 180 

Before Rotation: P ( x , y )
After Rotation: P’ ( -x, -y )

Let us say, for example, we have the following points, and we must rotate each point by 180 degrees. Identify the new position of each point.

( a ) E ( 4, 5 )
( b ) F ( -3, 6 )
( c ) G ( 9, -1 )
( d ) H ( -8, -2 )

If we rotate each point by 180° with respect to the origin, we must move based on the formula ( x, y ) ( -x, -y ).
The new position of point E ( 4, 5 ), when rotated by 180 degrees clockwise or counterclockwise, is E’ ( -4, -5 ).
The new position of point F ( -3, 6 ) when rotated by 180 degrees clockwise or counterclockwise is F’ ( 3, -6 ).
The new position of point G ( 9, -1 ) when rotated by 180 degrees clockwise or counterclockwise is G’ ( -9, 1 ).
The new position of point H ( -8, -2 ), when rotated by 180 degrees clockwise or counterclockwise, is H’ ( 8, 2 ).

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