# Lesson 1

Rigid Transformations in the Plane

- Let’s try transformations with coordinates.

### Problem 1

Reflect triangle \(ABC\) over the line \(x=\text-3\).

Translate the image by the directed line segment from \((0,0)\) to \((4,1)\).

What are the coordinates of the vertices in the final image?

### Problem 2

Three line segments form the letter N. Rotate the letter N counterclockwise around the midpoint of segment \(BC\) by 180 degrees. Describe the result.

### Problem 3

Triangle \(ABC\) has coordinates \(A = (1, 3), B = (2,0),\) and \(C = (4,1).\) The image of this triangle after a sequence of transformations is triangle \(A’B’C’\) where \(A’ = (\text-5,\text-3), B’ = (\text-4,0),\) and \(C’ = (\text-2,\text-1).\)

Write a sequence of transformations that takes triangle \(ABC\) to triangle \(A’B’C’\).

### Problem 4

Prove triangle \(ABC\) is congruent to triangle \(DEF\).

### Problem 5

The density of water is 1 gram per cm^{3}. An object floats in water if its density is less than water’s density, and it sinks if its density is greater than water’s. Will a 1.17 gram diamond in the shape of a pyramid whose base has area 2 cm^{2} and whose height is 0.5 centimeters sink or float? Explain your reasoning.

### Problem 6

*Technology required*. An oblique cylinder with a base of radius 2 units is shown. The top of the cylinder can be obtained by translating the base by the directed line segment \(AB\) which has length 16 units. The segment \(AB\) forms a \(30^\circ\) angle with the plane of the base. What is the volume of the cylinder?

### Problem 7

This design began from the construction of an equilateral triangle. Record at least 3 rigid transformations (rotation, reflection, translation) you see in the design.