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## High school geometry

### Course: High school geometry>Unit 1

Lesson 4: Rotations

# Rotating shapes about the origin by multiples of 90°

Learn how to draw the image of a given shape under a given rotation about the origin by any multiple of 90°.

## Introduction

In this article we will practice the art of rotating shapes. Mathematically speaking, we will learn how to draw the image of a given shape under a given rotation.
This article focuses on rotations by multiples of 90, degrees, both positive (counterclockwise) and negative (clockwise).

## Part 1: Rotating points by $90^\circ$90, degrees, $180^\circ$180, degrees, and $-90^\circ$minus, 90, degrees

### Let's study an example problem

We want to find the image A, prime of the point A, left parenthesis, 3, comma, 4, right parenthesis under a rotation by 90, degrees about the origin.
Let's start by visualizing the problem. Positive rotations are counterclockwise, so our rotation will look something like this:
A blank coordinate plane with a line segment where its endpoints are at the origin and a point at three, four labeled A. The point is rotated counter clockwise ninety degrees so that A prime is now in the second quadrant.
Cool, we estimated A, prime visually. But now we need to find exact coordinates. There are two ways to do this.

### Solution method 1: The visual approach

We can imagine a rectangle that has one vertex at the origin and the opposite vertex at A.
A coordinate plane with a rectangle with vertices at the origin, zero, four, three, zero, and three, four which is labeled A. The x- and y- axes scale by one.
A rotation by 90, degrees is like tipping the rectangle on its side:
A coordinate plane with a pre image rectangle with vertices at the origin, zero, four, three, zero, and three, four which is labeled A. The x- and y- axes scale by one. The rectangle is rotated ninety degrees to form the image of a rectangle with vertices at the origin, zero, three, negative four, zero, and negative four, three which is labeled A prime.
Now we see that the image of A, left parenthesis, 3, comma, 4, right parenthesis under the rotation is A, prime, left parenthesis, minus, 4, comma, 3, right parenthesis.
Notice it's easier to rotate the points that lie on the axes, and these help us find the image of A:
Pointleft parenthesis, 3, comma, 0, right parenthesisleft parenthesis, 0, comma, 4, right parenthesisleft parenthesis, 3, comma, 4, right parenthesis
Imageleft parenthesis, 0, comma, 3, right parenthesisleft parenthesis, minus, 4, comma, 0, right parenthesisleft parenthesis, minus, 4, comma, 3, right parenthesis

### Solution method 2: The algebraic approach

Let's take a closer look at points A and A, prime:
Pointx-coordinatey-coordinate
Astart color #01a995, 3, end color #01a995start color #aa87ff, 4, end color #aa87ff
A, primeminus, start color #aa87ff, 4, end color #aa87ffstart color #01a995, 3, end color #01a995
Notice an interesting phenomenon: The x-coordinate of A became the y-coordinate of A, prime, and the opposite of the y-coordinate of A became the x-coordinate of A, prime.
We can represent this mathematically as follows:
R, start subscript, left parenthesis, 0, comma, 0, right parenthesis, comma, 90, degrees, end subscript, left parenthesis, start color #01a995, x, end color #01a995, comma, start color #aa87ff, y, end color #aa87ff, right parenthesis, equals, left parenthesis, minus, start color #aa87ff, y, end color #aa87ff, comma, start color #01a995, x, end color #01a995, right parenthesis
It turns out that this is true for any point, not just our A. Here are a few more examples:
A coordinate plane with three pre image points at eight, negative one, negative three, four, and negative three, negative six. They are rotated counter clockwise to form the image points at one, eight, negative four, negative three, and six, negative three respectively. The x- and y- axes scale by one.
Furthermore, it turns out that rotations by 180, degrees or minus, 90, degrees follow similar patterns:
R, start subscript, left parenthesis, 0, comma, 0, right parenthesis, comma, 180, degrees, end subscript, left parenthesis, start color #01a995, x, end color #01a995, comma, start color #aa87ff, y, end color #aa87ff, right parenthesis, equals, left parenthesis, minus, start color #01a995, x, end color #01a995, comma, minus, start color #aa87ff, y, end color #aa87ff, right parenthesis
R, start subscript, left parenthesis, 0, comma, 0, right parenthesis, comma, minus, 90, degrees, end subscript, left parenthesis, start color #01a995, x, end color #01a995, comma, start color #aa87ff, y, end color #aa87ff, right parenthesis, equals, left parenthesis, start color #aa87ff, y, end color #aa87ff, comma, minus, start color #01a995, x, end color #01a995, right parenthesis
We can use these to rotate any point we want by plugging its coordinates in the appropriate equation.

#### Problem 1

Draw the image of B, left parenthesis, minus, 7, comma, minus, 3, right parenthesis under the rotation R, start subscript, left parenthesis, 0, comma, 0, right parenthesis, comma, minus, 90, degrees, end subscript.

#### Problem 2

Draw the image of C, left parenthesis, 5, comma, minus, 6, right parenthesis under the rotation R, start subscript, left parenthesis, 0, comma, 0, right parenthesis, comma, 180, degrees, end subscript.

### Graphical method vs. algebraic method

In general, everyone is free to choose which of the two methods to use. Different strokes for different folks!
The algebraic method takes less work and less time, but you need to remember those patterns. The graphical method is always at your disposal, but it might take you longer to solve.

## Part 2: Extending to any multiple of $90^\circ$90, degrees

### Let's study an example problem

We want to find the image D, prime of the point D, left parenthesis, minus, 5, comma, 4, right parenthesis under a rotation by 270, degrees about the origin.

### Solution

Since rotating by 270, degrees is the same as rotating by 90, degrees three times, we can solve this graphically by performing three consecutive 90, degrees rotations:
A coordinate plane with a pre image rectangle with vertices at the origin, zero, four, negative five, zero, and negative five, four which is labeled D. The x- and y- axes scale by one. The rectangle is rotated ninety degrees to form the image of a rectangle with vertices at the origin, zero, negative five, negative four, zero, and negative four, negative five. The rectangle is rotated ninety degrees again to form the image of a rectangle with vertices at the origin, zero, negative four, five, zero, and five, negative four. The rectangle is rotated a third time ninety degrees to form the image of a rectangle with vertices at the origin, zero, five, four, zero, and four, five which is labeled D prime.
But wait! We could just rotate by minus, 90, degrees instead of 270, degrees. These rotations are equivalent. Check it out:
A coordinate plane with a pre image rectangle with vertices at the origin, zero, four, negative five, zero, and negative five, four which is labeled D. The x- and y- axes scale by one. The rectangle is rotated ninety degrees clockwise to form the image of a rectangle with vertices at the origin, zero, five, four, zero, and four, five which is labeled D prime.
For the same reason, we can also use the pattern R, start subscript, left parenthesis, 0, comma, 0, right parenthesis, comma, minus, 90, degrees, end subscript, left parenthesis, start color #01a995, x, end color #01a995, comma, start color #aa87ff, y, end color #aa87ff, right parenthesis, equals, left parenthesis, start color #aa87ff, y, end color #aa87ff, comma, minus, start color #01a995, x, end color #01a995, right parenthesis:
R, start subscript, left parenthesis, 0, comma, 0, right parenthesis, comma, 270, degrees, end subscript, left parenthesis, minus, 5, comma, 4, right parenthesis, equals, left parenthesis, 4, comma, 5, right parenthesis

### Let's study one more example problem

We want to find the image of left parenthesis, minus, 9, comma, minus, 7, right parenthesis under a rotation by 810, degrees about the origin.

### Solution

A rotation by 810, degrees is the same as two consecutive rotations by 360, degrees followed by a rotation by 90, degrees (because 810, equals, 2, dot, 360, plus, 90).
A rotation by 360, degrees maps every point onto itself. In other words, it doesn't change anything.
So a rotation by 810, degrees is the same as a rotation by 90, degrees. Therefore, we can simply use the pattern R, start subscript, left parenthesis, 0, comma, 0, right parenthesis, comma, 90, degrees, end subscript, left parenthesis, start color #01a995, x, end color #01a995, comma, start color #aa87ff, y, end color #aa87ff, right parenthesis, equals, left parenthesis, minus, start color #aa87ff, y, end color #aa87ff, comma, start color #01a995, x, end color #01a995, right parenthesis:
R, start subscript, left parenthesis, 0, comma, 0, right parenthesis, comma, 810, degrees, end subscript, left parenthesis, minus, 9, comma, minus, 7, right parenthesis, equals, left parenthesis, 7, comma, minus, 9, right parenthesis

#### Problem 1

Draw the image of E, left parenthesis, 8, comma, 6, right parenthesis under the rotation R, start subscript, left parenthesis, 0, comma, 0, right parenthesis, comma, minus, 270, degrees, end subscript.

#### Problem 2

Which rotation is equivalent to the rotation R, start subscript, left parenthesis, 0, comma, 0, right parenthesis, comma, minus, 990, degrees, end subscript?

## Part 3: Rotating polygons

### Let's study an example problem

Consider quadrilateral D, E, F, G drawn below. Let's draw its image, D, prime, E, prime, F, prime, G, prime, under the rotation R, start subscript, left parenthesis, 0, comma, 0, right parenthesis, comma, 270, degrees, end subscript.
A cooordinate plane with a quadrilateral with vertices D at five, five, E at seven, six, F at eight, negative two, and G at two, negative two. The x- and y- axes scale by one.

### Solution

Similar to translations, when we rotate a polygon, all we need is to perform the rotation on all of the vertices, and then we can connect the images of the vertices to find the image of the polygon.
A cooordinate plane with a pre image quadrilateral with vertices D at five, five, E at seven, six, F at eight, negative two, and G at two, negative two. The x- and y- axes scale by one. It is rotated two hundred seventy degrees counter clockwise to form the image of the quadrilateral with vertices D prime at five, negative five, E prime at six, negative seven, F prime at negative two, negative eight, and G prime at negative two, negative two.