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://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=20240718T033732Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=c006e159db013999ea3015aab5c1c24139e0f097e2c7436dbd37b6c565046213)
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://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=20240718T033732Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=2d3338b41ef6f490b70b985a18781c5bb1b20647d9d772af99f7fcfeff896efc)
- 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://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=20240718T033732Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=371bd1b7c8414100a32fbfcb076e36de884c00ccc286935c8935f5616aebe85a)
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 \(Z\).