Lesson 9
Composing Figures
Let’s use reasoning about rigid transformations to find measurements without measuring.
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://staging-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=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T132618Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=ad988e4e5cdd6bcb8eacbb701d691332eac7a5b2cceb2173f044e65fe109de8a)
Describe a sequence of translations, rotations, and reflections that take the lower left triangle to the upper right triangle.
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://staging-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=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T132618Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=41422cea9c64b4c895adeb13ee7c6577083021b6d5a3011f6321fc4a3e1f6fdf)
- 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.
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 \(p\).
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?
Problem 5
Here is triangle \(XYZ\):
![Triangle X Y Z appears isosceles, with Z Y vertical and Z X congruent to Y X.](https://staging-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=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T132618Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=b0b8bff94eac415a7852fc3e8f7d651ca571be4949a46be574dbd9f90544c616)
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 \(Z\).