# Lesson 3

Types of Transformations

- Let’s analyze transformations that produce congruent and similar figures.

### Problem 1

Complete the table and determine the rule for this transformation.

input | output |
---|---|

\((2,\text-3)\) | \((\text-3,2)\) |

\((4,5)\) | \((5,4)\) |

\((\rule{.5cm}{0.4pt},4)\) | \((4,0)\) |

\((1,6)\) | |

\((\text-1,\text-2)\) | |

\((x,y)\) |

### Problem 2

Write a rule that describes this transformation.

original figure | image |
---|---|

\((5,1)\) | \((2,\text-1)\) |

\((\text-3,4)\) | \((\text-6,\text-4)\) |

\((1,\text-2)\) | \((\text-2,2)\) |

\((\text-1,\text-4)\) | \((\text-4,4)\) |

### Problem 3

Select **all** the transformations that produce congruent images.

dilation

horizontal stretch

reflection

rotation

translation

### Problem 4

Here are some transformation rules. For each transformation, first predict what the image of triangle \(ABC\) will look like. Then compute the coordinates of the image and draw it.

- \((x,y) \rightarrow (x-4,y-1)\)
- \((x,y) \rightarrow (y,x)\)
- \((x,y) \rightarrow (1.5x,1.5y)\)

### Problem 5

A cylinder has radius 3 inches and height 5 inches. A cone has the same radius and height.

- Find the volume of the cylinder.
- Find the volume of the cone.
- What fraction of the cylinder’s volume is the cone’s volume?

### Problem 6

Reflect triangle \(ABC\) over the line \(x=\text-2\). Call this new triangle \(A’B’C’\). Then reflect triangle \(A’B’C’\) over the line \(x=0\). Call the resulting triangle \(A''B''C''\).

Describe a single transformation that takes \(ABC\) to \(A''B''C''\).

### Problem 7

In the construction, \(A\) is the center of one circle, and \(B\) is the center of the other.

Explain why segments \(AC\), \(BC\), \(AD\), \(BD\), and \(AB\) have the same length.