Lesson 9
Composing Figures
Problem 1
Here is the design for the flag of Trinidad and Tobago.
![The flag of Trinidad and Tobago: a red rectangle with a black stripe outlined with narrow white stripe from upper left corner to lower right corner.](https://cms-im.s3.amazonaws.com/qoBoVgBGeMFgKamB66rUe9Ee?response-content-disposition=inline%3B%20filename%3D%228-8.1.B.PP.Image.08.3.png%22%3B%20filename%2A%3DUTF-8%27%278-8.1.B.PP.Image.08.3.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240718%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240718T032355Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=34a5b9db5a88365981a85fdd11b1ff01969043913b1b17047385e6ba31ae46db)
Describe a sequence of translations, rotations, and reflections that take the lower left triangle to the upper right triangle.
Solution
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Problem 2
Here is a picture of an older version of the flag of Great Britain. There is a rigid transformation that takes Triangle 1 to Triangle 2, another that takes Triangle 1 to Triangle 3, and another that takes Triangle 1 to Triangle 4.
![An image of an older version of the flag of Great Britain.](https://cms-im.s3.amazonaws.com/FbJ7kP2SosZWMeDEALpfRbSM?response-content-disposition=inline%3B%20filename%3D%228-8.1.B.PP.Image.08.4.png%22%3B%20filename%2A%3DUTF-8%27%278-8.1.B.PP.Image.08.4.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240718%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240718T032355Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=e2e8d803604d2681336284e2f2097b7cb90ed9dfd4775472defd3e90b48d3e49)
- Measure the lengths of the sides in Triangles 1 and 2. What do you notice?
- What are the side lengths of Triangle 3? Explain how you know.
- Do all eight triangles in the flag have the same area? Explain how you know.
Solution
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Problem 3
- Which of the lines in the picture is parallel to line \(\ell\)? Explain how you know.
- Explain how to translate, rotate or reflect line \(\ell\) to obtain line \(k\).
- Explain how to translate, rotate or reflect line \(\ell\) to obtain line \(p\).
Solution
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(From Unit 1, Lesson 8.)Problem 4
Point \(A\) has coordinates \((3,4)\). After a translation 4 units left, a reflection across the \(x\)-axis, and a translation 2 units down, what are the coordinates of the image?
Solution
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(From Unit 1, Lesson 5.)Problem 5
Here is triangle \(XYZ\):
![Triangle X Y Z appears isosceles, with Z Y vertical and Z X congruent to Y X.](https://cms-im.s3.amazonaws.com/3V8rsSgabnvFUuPAP4PpGVHg?response-content-disposition=inline%3B%20filename%3D%228-8.1.B8.newPP.03.png%22%3B%20filename%2A%3DUTF-8%27%278-8.1.B8.newPP.03.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240718%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240718T032355Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=7711ea5e6869168266e73d1da3c9bc12834e81948bb75f522df4aa5229a53381)
Draw these three rotations of triangle \(XYZ\) together.
- Rotate triangle \(XYZ\) 90 degrees clockwise around \(Z\).
- Rotate triangle \(XYZ\) 180 degrees around \(Z\).
- Rotate triangle \(XYZ\) 270 degrees clockwise around \(Z\).
Solution
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(From Unit 1, Lesson 7.)