Geometry 9-4 Complete Lesson: Compositions of Isometries

By Matt Richardson

Last updated almost 3 years ago

Note from the author:

A complete formative lesson with embedded slideshow, mini lecture screencasts, checks for understanding, practice items, mixed review, and reflection. I create these assignments to supplement each lesson of Pearson's Common Core Edition Algebra 1, Algebra 2, and Geometry courses. See also mathquest.net and twitter.com/mathquestEDU.

Solve It! The blue E is a horizontal translation of the red E. How can you use two reflections, one right after the other, to move the red E to the position of the blue E?

G.CO.5

Solve It! Draw the two lines of reflection you used on the image on the canvas.

G.CO.5

Problem 1 Got It? Lines l and m are copied from Problem 1. Draw J between l and m. Draw the image of the following composition of reflections in blue or green.

G.CO.5

G.CO.6

Problem 1 Got It? What is the distance of the resulting translation in the previous item?

Problem 1 Got It? Reasoning: Use the results of the previous items and Problem 1. Make a conjecture about the distance of any translation that is the result of a composition of reflections across two parallel lines.

G.CO.5

G.CO.6

Problem 2 Got It? Draw the following composition of reflections on the canvas.

G.CO.5

G.CO.6

Problem 2 Got It? What is the center for the resulting rotation?

G.CO.5

G.CO.6

Problem 2 Got It? What is the angle of rotation for the resulting rotation?

G.CO.5

G.CO.6

Problem 2 Got It? Reasoning: Use the results of the previous items and Problem 2. Make a conjecture about the center of rotation and the angle of rotation for any rotation that is the result of any composition of reflections across two intersecting lines.

Problem 3 Got It? Graph ΔTEX in black. Graph the image ΔT'E'X' for the glide reflection in green or blue. Be sure to include relevant graph detail.

G.CO.5

G.CO.6

For each diagram, sketch the image of Z reflected across line a, then across line b. Use green or blue.

G.CO.5

G.CO.6

G.CO.5

G.CO.6

Error Analysis: You reflect ∆DEF first across line m and then across line n. Your friend says you can get the same result by reflecting ∆DEF first across line n and then across line m. Explain your friend's error.

G.CO.5

Review Lesson 9-3: Draw ∆ABC in black and its image for the rotation described in blue or green. Label all vertices.

G.CO.5

G.CO.6

Review Lesson 9-3: Draw ∆ABC in black and its image for the rotation described in blue or green. Label all vertices.

G.CO.5

G.CO.6

Review Lesson 5-5: Identify the two statements that contradict each other.

△ABC is a right triangle.
△ABC is equiangular.
△ABC is isosceles.

Statements that contradict each other

Review Lesson 5-5: Identify the two statements that contradict each other.

In right △ABC, m∠B = 90.
In right △ABC, m∠A = 80.
In right △ABC, m∠C = 90.

Statements that contradict each other

Review Lesson 4-2: Determine whether the triangles in each pair must be congruent. If so, name the postulate of theorem that justifies your answer.

Pair 1
Pair 2
Pair 3

Not necessarily congruent
SSS
ASA

Vocabulary Review: Match the pairs of corresponding sides.

segment JQ
segment KQ
segment J'K'

segment J'Q'
segment JK
segment K'Q'

Vocabulary Review: Complete each sentence.

reflection
∠K'
translation
∠J
∠Q
rotation
∠J'
dilation

The image of ∠J is __?__.
The transformation △JKQ → △J'K'Q' is a __?__.

Use Your Vocabulary: Complete each sentence with image or preimage.

image
preimage

In a transformation, the original figure is the __?__.
The resulting figure is the __?__.

Geometry 9-4 Complete Lesson: Compositions of Isometries

Solve It! The blue E is a horizontal translation of the red E. How can you use two reflections, one right after the other, to move the red E to the position of the blue E?

Solve It! Draw the two lines of reflection you used on the image on the canvas.

Problem 1 Got It? Lines l and m are copied from Problem 1. Draw J between l and m. Draw the image of the following composition of reflections in blue or green.

Problem 1 Got It? What is the distance of the resulting translation in the previous item?

Problem 1 Got It? Reasoning: Use the results of the previous items and Problem 1. Make a conjecture about the distance of any translation that is the result of a composition of reflections across two parallel lines.

Problem 2 Got It? Draw the following composition of reflections on the canvas.

Problem 2 Got It? What is the center for the resulting rotation?

Problem 2 Got It? What is the angle of rotation for the resulting rotation?

Problem 2 Got It? Reasoning: Use the results of the previous items and Problem 2. Make a conjecture about the center of rotation and the angle of rotation for any rotation that is the result of any composition of reflections across two intersecting lines.

Problem 3 Got It? Graph ΔTEX in black. Graph the image ΔT'E'X' for the glide reflection in green or blue. Be sure to include relevant graph detail.

For each diagram, sketch the image of Z reflected across line a, then across line b. Use green or blue.

Error Analysis: You reflect ∆DEF first across line m and then across line n. Your friend says you can get the same result by reflecting ∆DEF first across line n and then across line m. Explain your friend's error.

Review Lesson 9-3: Draw ∆ABC in black and its image for the rotation described in blue or green. Label all vertices.

Review Lesson 9-3: Draw ∆ABC in black and its image for the rotation described in blue or green. Label all vertices.

Review Lesson 5-5: Identify the two statements that contradict each other.

Review Lesson 5-5: Identify the two statements that contradict each other.

Review Lesson 4-2: Determine whether the triangles in each pair must be congruent. If so, name the postulate of theorem that justifies your answer.

Vocabulary Review: Match the pairs of corresponding sides.

Vocabulary Review: Complete each sentence.

Use Your Vocabulary: Complete each sentence with image or preimage.

Use Your Vocabulary: Which type of transformation maps each (x, y) to (x - 8, y + 2)?

Reflection: Math Success