Lesson 6
Similarity
Let’s explore similar figures.
Problem 1
Each diagram has a pair of figures, one larger than the other. For each pair, show that the two figures are similar by identifying a sequence of translations, rotations, reflections, and dilations that takes the smaller figure to the larger one.
![Coordinate plane, x 0 to 4, y 0 to 6. Line through point A, the origin, C at 1 comma 2, E at 3 comma 6. Segments connect A, & C to point B at 1 comma 0. Segments connect E & C to point F at 3 comma 2.](https://staging-cms-im.s3.amazonaws.com/5wXB9BTeAqmgr9KTQ3aLsAFM?response-content-disposition=inline%3B%20filename%3D%228-8.2.B.PP.Image.03.png%22%3B%20filename%2A%3DUTF-8%27%278-8.2.B.PP.Image.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=20240722T143227Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=3426a1e463d543aa1f7894cb2dfee292f22b61de56c3a56078c769207bbcd42c)
![Two triangles on a circular grid. Ask for further assistance.](https://staging-cms-im.s3.amazonaws.com/gDtbrvqDBWtUs4cQacqJkKm4?response-content-disposition=inline%3B%20filename%3D%228-8.2.B.PP.Image.04.png%22%3B%20filename%2A%3DUTF-8%27%278-8.2.B.PP.Image.04.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=20240722T143227Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=8c95bce5b6bbdf07b2eac710c7aff4e00d4ad53ef6d2e8b722b94ef71ae19ed2)
Problem 2
Here are two similar polygons.
Measure the side lengths and angles of each polygon. What do you notice?
![figure ABCD and figure EFGH on a grid.](https://staging-cms-im.s3.amazonaws.com/3EtfbbrKxN68Z5KAzaexR6Bh?response-content-disposition=inline%3B%20filename%3D%228-8.2.B.PP.Image.02.png%22%3B%20filename%2A%3DUTF-8%27%278-8.2.B.PP.Image.02.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=20240722T143227Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=1bedc5dcf5151ee9b2191980d62bd6787a75820467af656752a5df1dbaf3bd96)
Problem 3
Each figure shows a pair of similar triangles, one contained in the other. For each pair, describe a point and a scale factor to use for a dilation moving the larger triangle to the smaller one. Use a measurement tool to find the scale factor.
![Triangles A, B C, and A prime B C prime. A prime lies on side A, B. C prime lies on side B C.](https://staging-cms-im.s3.amazonaws.com/JS2VrKd4kvVqgTfqpttfjWYA?response-content-disposition=inline%3B%20filename%3D%228-8.2.B.PP.Image.05.png%22%3B%20filename%2A%3DUTF-8%27%278-8.2.B.PP.Image.05.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=20240722T143227Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=3862201664eea633e7ff56e29cb8c044eb0579d96af66ad709b3945214ed0998)
![Triangles A, B C, and A, B prime C prime. B prime lies on side A, B. C prime lies on side A, C.](https://staging-cms-im.s3.amazonaws.com/tY3T5tQJZK3BdYysFQKBSMb4?response-content-disposition=inline%3B%20filename%3D%228-8.2.B.PP.Image.06.1.png%22%3B%20filename%2A%3DUTF-8%27%278-8.2.B.PP.Image.06.1.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=20240722T143227Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=80e6633f4312d2da5c71cb2cebde94ebd6ef0961bd282344a87f85adaf4e6d10)