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?

triangle ABC graphed. A = -5 comma 2, B = -6 comma -1, C = -3 comma 0.

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.

The letter N, formed by 3 line segments. 4 points on the endpoints of the segments. Starting at the bottom left, A. Moving upward, B. Slanting down and to the right, C. Moving upward, D.
(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 \(A’B’C’\).

Two triangles in coordinate plane.

Problem 4

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

Triangles ABC and DEF. A at origin, B at 2 comma 0, C at 3 comma 4, D at 3 comma -1, E at 3 comma 1, F at -1 comma 2. 


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. 

(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?

Oblique cylinder. Endpoints of slant length labeled A, at bottom and B, at top.
(From Unit 5, Lesson 11.)

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.

Equilateral triangle with three small congruent circles.
(From Unit 1, Lesson 22.)