Lesson 11
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%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T233504Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=6e7886cd42dd605fc1881eb67e2e034410840218f094ee97bd3ed6f40b1a84a5)
![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%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T233504Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=5522bc945a8deb3d2993a6a7813b1c29e2650b83caf9e1140872789b0fec47e9)
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%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T233504Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=d15662a138c26331275a13c565c46238af0734d30fcabf04ecce28ebea8c10ee)
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%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T233504Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=84d4607389e56d0dfa574f0aab6cb0eef332b541a2119e81e7eb5015255ececd)
![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%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T233504Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=87a9934221ab44e287635de08068731683ed937eab50e4c7ea366b4df9386bb4)
Problem 4
Describe a sequence of translations, rotations, and reflections that takes Polygon P to Polygon Q.
![two of the same figure on a square grid in different orientations and position](https://staging-cms-im.s3.amazonaws.com/JLHRzfYA4dQuZvDmzCTNJA7Y?response-content-disposition=inline%3B%20filename%3D%228-8.1.A.PP.Image.18b.png%22%3B%20filename%2A%3DUTF-8%27%278-8.1.A.PP.Image.18b.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T233504Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=4042928b36d6acb0cd1edbc85196590b80091ba325bd3c50bd82b814f2f9321d)
Problem 5
- Draw the translated image of \(ABCDE\) so that vertex \(C\) moves to \(C’\). Tracing paper may be useful.
- Draw the reflected image of Pentagon \(ABCDE\) with line of reflection \(\ell\). Tracing paper may be useful.
- Draw the rotation of Pentagon \(ABCDE\) around \(C\) clockwise by an angle of 150 degrees. Tracing paper and a protractor may be useful.