Lesson 1

Rigid Transformations in the Plane

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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.

Solution

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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.

Solution

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(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.

Solution

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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. 

 

Solution

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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

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(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.

Solution

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(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.

Solution

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(From Unit 1, Lesson 22.)