# Lesson 1

Rigid Transformations in the Plane

### 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.

### Solution

(From Unit 1, Lesson 14.)

### 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 cm3. 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 cm2 and whose height is 0.5 centimeters sink or float? Explain your reasoning.

### Solution

(From Unit 5, Lesson 17.)

### 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?