# Determining rotations

Learn how to determine which rotation brings one given shape to another given shape.
In each of these practice problems, a rotation has already occurred. We are given the source (the shape before the rotation), the image (the shape after the rotation), and the center of rotation. From there, we are asked to find the angle of rotation.
Figuring out the angle of rotation without any help is pretty difficult. Thankfully, we can use our rotation tool to try out different angles until we find one that works.
If you find the problems too confusing, we encourage you to click on the "I need help!" button.

## Problem 1

Map quadrilateral start color blue, A, B, C, D, end color blue onto quadrilateral start color maroonD, A, prime, B, prime, C, prime, D, prime, end color maroonD using a rotation about the origin left parenthesis, 0, comma, 0, right parenthesis.

### First try: A $180^\circ$180, degree rotation

The quadrilaterals seem to be on almost opposite sides of the origin, so we can start by rotating start color blueD, A, B, C, D, end color blueD by 180, degree and seeing where it lands.
We can see that the image moved past start color maroonD, A, prime, B, prime, C, prime, D, prime, end color maroonD. This means that a degree of 180, degree is too large for what we need.
It's important to remember that a rotation by a positive angle is counterclockwise. For this reason, a degree of 180, degree is too large and not too small.

### Second try: A $160^\circ$160, degree rotation

Since 180, degree is too large, we should try a smaller degree of rotation. 160, degree seems like a good candidate: smaller than 180, degree but not too small.
We can see that the image falls short of reaching start color maroonD, A, prime, B, prime, C, prime, D, prime, end color maroonD. This means that a degree of 160, degree is too small for what we need.

### Third try: A $170^\circ$170, degree rotation

So 180, degree was too large and 160, degree was too small...
Let's try the middle value, 170, degree.
And we have a perfect match! Third time's the charm!

## Problem 2

Map start color blue, triangle, E, F, G, end color blue onto start color maroonD, triangle, E, prime, F, prime, G, prime, end color maroonD using a rotation about the point left parenthesis, 3, comma, minus, 3, right parenthesis.

### First try: A $-90^\circ$minus, 90, degree rotation

By quickly examining the drawing, it seems that start color maroonD, triangle, E, prime, F, prime, G, prime, end color maroonD is the result of a rotation, centered at left parenthesis, 3, comma, minus, 3, right parenthesis, by about 90, degree in the clockwise direction.
We should always remember that the clockwise direction is negative, so our first try is a rotation by minus, 90, degree.
The image moved past start color maroonD, triangle, E, prime, F, prime, G, prime, end color maroonD. This means that a degree of 90, degree clockwise is too large.

### Second try: A $-70^\circ$minus, 70, degree rotation

Similar to the first example, let's now try an angle smaller than 90, degree. There are many possibilities, but we will go with 70, degree. Again, remember that the rotation is by minus, 70, degree and not by plus, 70, degree.
The image falls short of triangle, E, prime, F, prime, G, prime, which means that a degree of 70, degree clockwise is too small.

### Third try: A $-75^\circ$minus, 75, degree rotation

We saw that an angle of 90, degree was too large and that an angle of 70, degree was too small.
Furthermore, we can see that a rotation by 70, degree got closer to start color maroonD, triangle, E, prime, F, prime, G, prime, end color maroonD than a rotation by 90, degree.
Therefore, instead of trying the middle value, 80, degree, we should try something closer to 70, degree. A natural candidate is 75, degree.
And we got it!