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High school geometry
Rotating shapes about the origin by multiples of 90°
CCSS.Math:
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, degrees, 180, degrees, and 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:
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 rotation by 90, degrees is like tipping the rectangle on its side:
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:
| Point | left parenthesis, 3, comma, 0, right parenthesis | left parenthesis, 0, comma, 4, right parenthesis | left parenthesis, 3, comma, 4, right parenthesis |
|---|---|---|---|
| Image | left parenthesis, 0, comma, 3, right parenthesis | left parenthesis, minus, 4, comma, 0, right parenthesis | left 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:
| Point | x-coordinate | y-coordinate |
|---|---|---|
| A | start color #01a995, 3, end color #01a995 | start color #aa87ff, 4, end color #aa87ff |
| A, prime | minus, start color #aa87ff, 4, end color #aa87ff | start 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:
It turns out that this is true for any point, not just our A. Here are a few more examples:
Furthermore, it turns out that rotations by 180, degrees or minus, 90, degrees follow similar patterns:
We can use these to rotate any point we want by plugging its coordinates in the appropriate equation.
Your turn!
Problem 1
Problem 2
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, 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:
But wait! We could just rotate by minus, 90, degrees instead of 270, degrees. These rotations are equivalent. Check it out:
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:
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:
Your turn!
Problem 1
Problem 2
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.
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.
Your turn!
Want to join the conversation?
im confused i dont get this(47 votes)
In case the algebraic method can help you:
Rotating by 90 degrees:
If you have a point on (2, 1) and rotate it by 90 degrees, it will end up at (-1, 2)
When you rotate by 90 degrees, you take your original X and Y, swap them, and make Y negative.
So from 0 degrees you take (x, y), swap them, and make y negative (-y, x) and then you have made a 90 degree rotation.
What if we rotate another 90 degrees? Same thing.
Our point is at (-1, 2) so when we rotate it 90 degrees, it will be at (-2, -1)
X and Y swaps, and Y becomes negative.
What about 90 degrees again? Same thing! But remember that a negative and a negative gives a positive so when we swap X and Y, and make Y negative, Y actually becomes positive.
Our point is as (-2, -1) so when we rotate it 90 degrees, it will be at (1, -2)
Another 90 degrees will bring us back where we started.
From (1, -2) to (2, 1)
Rotating by -90 degrees:
If you understand everything so far, then rotating by -90 degrees should be no issue for you.
We do the same thing, except X becomes a negative instead of Y.
So from (x, y) to (y, -x)
Rotating by 180 degrees:
If you have a point on (2, 1) and rotate it by 180 degrees, it will end up at (-2, -1)
When you rotate by 180 degrees, you take your original x and y, and make them negative.
So from 0 degrees you take (x, y) and make them negative (-x, -y) and then you've made a 180 degree rotation.
Remember! A negative and a negative gives a positive! So if we rotate another 180 degrees we go from (-2, -1) to (2, 1)
And if we have another point like (-3, 2) and rotate it 180 degrees, it will end up on (3, -2)(34 votes)
Anyone have any tips for visualization? I am having a really hard time seeing a triangle and where the point should go in my head. Any suggestions are appreciated very much!
Thank you,
Clarebugg(16 votes)
you can try thinking of it as a mountain
the bottom is the 2 points that stretch out and the top is the peak.
there are many different ways to think about it.
you are problem-solving by trying to visualize.(15 votes)
For the first problem, why does the solution say a rotation of 90 degrees when its asking for -270(5 votes)- A rotation of 90 degrees is the same thing as -270 degrees. They are called coterminal angles. A full way around a circle is 360 degrees, right? So you can find an angle by adding 360. -270 + 360 = 90
Hope this helps!(29 votes)
how do you solve for -180(5 votes)
That's the same thing as 180 degrees so just rotate 180 degrees either clockwise or anti-clockwise.(14 votes)
This is very confusing because of the whole counter clock wise and clock wise, its almost as if its backwards, is there any easy way to this?(10 votes)- Positive is counterclockwise and negative is clockwise(3 votes)
- Is it possible to use two different methods at once to solve an equation?(4 votes)
Yeah, you can check your answer with those. Also, it is helpful to do two ways.(6 votes)
- I made a summary of the formulas if anyone dont want to use the visualize method.
R=rotate
R(0,0)90°/-270°[counterclock wise/clockwise](x,y)➔(-y,x)
R(0,0)180°/-180°[counterclock wise/clock wise](x,y)➔(-x,-y)
R(0,0)-90°/270°[clock wise/counterclock wise](x,y)➔(y,-x)
R(0,0)360°/-360°(x,y)=(x,y)
In all honesty, there are only two that needed to memorize. R 90°/-270°and R -90°/270°since depends on the direction it rotates the x and y value sometimes turns into negative, but 100% of the time the x and y swaps. The rest of two you just need to know how it works, R 180°/-180 is just reflected through origin so you dont have to swap x and y, all you have to do is add - sign to them. And R360°/-360° is just rotating back to where it was, so nothing is changed.(8 votes) 
is -180 the same as 180?(5 votes)
Not quite the same, but they end at the same point. 180 degrees rotates the point counterclockwise and -180 degrees rotates the point clockwise.(6 votes)
- R(0,0)
why do you have to put two zeros inside R() and what does it mean?(6 votes)
The two 0s is the center of the of the grid, or the origin(2 votes)
How do you figure out what -990 is equivalent to? Is there a trick? Equation to figure this out?(4 votes)- The trick is to divide by 360 (full circle) then subtract the whole number and re-multiply the decimal times 360. So -990/360 is -2.75, add 2 to get -.75 * 360 = -270. OR if you add 3, you end up with .25 * 360 = 90. Both 90 and -270 are the same angle on the unit circle.(4 votes)