INTRO

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A rotation is a type of rigid transformation, which means it changes the position or orientation of an image without changing its size or shape.

When we simplify, we can see that the counterclockwise and clockwise rotations are just the reverse of each other. So there are really only 3 rotation formulas we need to remember.

A

__rotat__ion does this by__rotat__ing an image a certain amount of degrees either clockwise ↻ or counterclockwise ↺.For rotations of $90_{∘}$, $180_{∘}$, and $270_{∘}$ in either direction

**around the origin**$(0, 0)$, there are formulas we can use to figure out the new points of an image after it has been rotated.Clockwise ↻ | Counterclockwise ↺ | |

$90_{∘}$ | $(x,y)→(y,−x)$ | $(x,y)→(−y,x)$ |

$180_{∘}$ | $(x,y)→(−x,−y)$ | $(x,y)→(−x,−y)$ |

$270_{∘}$ | $(x,y)→(−y,x)$ | $(x,y)→(y,−x)$ |

CALCULATOR

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## Rotation Calculator

### Step 1. Identify the center of rotation.

KEY STEPS

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## How to Perform Rotations

### Step 1. Identify the center of rotation.

### Step 2. Identify the original points.

$original points=(x_{1},y_{1}),(x_{2},y_{2}),...,(x_{n},y_{n})$

### Step 3. Identify the angle and direction of rotation.

Direction:Angle of Rotation:

### Step 4. Identify the formula that matches the rotation.

When we rotate $90_{∘}$ counterclockwise:

- The $x$ axis lines up with the $y$ arrow pointing in the
**negative**direction, so the new $x$ value is the**negative**of the old $y$ value. - The $y$ axis lines up with the $x$ arrow pointing in the positive direction, so the new $y$ value is the old $x$ value.

### Step 5. Apply the formula to each original point to get the new points.

Original Point | New Point |

$(x_{1},y_{1})$ | $(−y_{1},x_{1})$ |

$(x_{2},y_{2})$ | $(−y_{2},x_{2})$ |

... | ... |

$(x_{n},y_{n})$ | $(−y_{n},x_{n})$ |

### Step 6. Plot the new points.

LESSON

— Rotations around the Origin

## Rotations around the Origin

We can use the following formulas for clockwise and counterclockwise rotations of $90_{∘}$, $180_{∘}$, and $270_{∘}$:Rotation | Formula |

$90_{∘}$ clockwise or $270_{∘}$ counterclockwise | $(x,y)→(y,−x)$ |

$180_{∘}$ clockwise or $180_{∘}$ counterclockwise | $(x,y)→(−x,−y)$ |

$270_{∘}$ clockwise or $90_{∘}$ counterclockwise | $(x,y)→(−y,x)$ |

**counter**clockwise (↺) means the

**opposite**of that.

### Clockwise ↻

### Counterclockwise ↺

Rotation | Original | New |

$90_{∘}$ clockwise or $270_{∘}$ counterclockwise | $(x,y)=(x_{original},y_{original})$ | $(y,−x)=(x_{new},y_{new})$ |

$180_{∘}$ clockwise or $180_{∘}$ counterclockwise | $(x,y)=(x_{original},y_{original})$ | $(−x,−y)=(x_{new},y_{new})$ |

$270_{∘}$ clockwise or $90_{∘}$ counterclockwise | $(x,y)=(x_{original},y_{original})$ | $(−y,x)=(x_{new},y_{new})$ |

PRACTICE

— Rotations around the Origin

## Practice: Rotations Around the Origin

Question 1 of 5: Rotate the following shape $90_{∘}$ clockwise around the origin.

### Step 1. Identify the center of rotation.

$center of rotation=$

$($$,$$)$

LESSON

— Rotations NOT around the Origin

## Rotations NOT around the Origin

Rotations that are not around the origin are a little trickier, but still 100% doable if we take it one step at a time.Let’s walk through some examples to help us remember the rotation formulas!PRACTICE

— Rotations NOT around the Origin

## Practice: Rotations NOT Around the Origin

Question 1 of 5: Rotate the following shape $90_{∘}$ clockwise around the point $(3,−2)$.

### Step 1. Identify the center of rotation.

$center of rotation=$

$($$,$$)$

CONCLUSION

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Nice work, look at you go! Thanks for checking out this lesson ☺️🙏. Where to next?

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