(a, b) x y r. For this problem, is a point in the - coordinate plane and is a positive number.

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Sherman Stephens
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1 Illustative G-C Simila cicles Alignments to Content Standads: G-C.A. Task (a, b) x y Fo this poblem, is a point in the - coodinate plane and is a positive numbe. a. Using a tanslation and a dilation, show how to tansfom the cicle with adius (a, b) (0, 0) centeed at into the cicle of adius centeed at. b. Explain how to use you wok in pat (a) to show that any two cicles ae simila. IM Commentay The standad G-C. asks us to pove that all cicles ae simila. This means that given any two cicles thee is a sequence of tansfomations of the plane (eflections, otations, tanslations, and dilations) tansfoming one to the othe. If we give the plane a coodinate system, then the cicles have equations and the tansfomations can be descibed explicitly with equations. The goal of this task is to wok on showing that all cicles ae simila using these two diffeent methods, the fist visual and the second algebaic. The teache may need to povide guidance, both with the geometic visualization and its tanslation into equations: this could be accomplished, fo example, by doing an example, that is by choosing explicit values fo a, b, and, pefeably whole numbes. Along the same lines, the teache could give the students two explicit equations of cicles and have them show that these cicles ae simila. This task is ideally suited fo goup wok whee students can shae ideas and insights and help one anothe put togethe the diffeent ideas equied to successfully complete this task.

2 Illustative Pat (a) of this poblem will equie MP (Make Sense of Poblems and Pesevee in Solving Them) as some thought needs to given about which tanslation and dilation to use as well as how to epesent these algebaically. Pat (b) povides an oppotunity fo students to engage in MP8 (Look Fo and Expess Regulaity in Repeated Reasoning) since one natual way to show this is to use the agument in pat (a) twice, that is conclude that any two cicles must be simila because they ae each simila to the unit cicle centeed at the oigin. This task includes an expeimental GeoGeba woksheet, with the intent that instuctos might use it to moe inteactively demonstate the elevant content mateial. The file should be consideed a daft vesion, and feedback on it in the comment section is highly encouaged, both in tems of suggestions fo impovement and fo ideas on using it effectively. The file can be un via the fee online application GeoGeba, o un locally if GeoGeba has been installed. Solutions Edit this solution Solution: Tansfomations This fist appoach to the poblem uses tansfomations of the plane to move the given cicle to the cicle of adius centeed at. a. To move the cente of the cicle to we need to tanslate by in the - diection and b y (0, 0) (0, 0) a x in the -diection. This is pictued below: 2

3 Illustative (0, 0) (0, 0) (0, 0) In ode to tansfom ou cicle of adius centeed at into a cicle of adius centeed at we can apply a dilation, with cente, with scale facto. This is pictued below (in this case so this is a contaction): > 3

4 Illustative The geen aows show whee selected points on the blue cicle of adius map to on the ed cicle of adius. C C 2 b. Pat (a) applies to any cicle in the plane. If and ae two cicles then by pat (a) C C 2 (0, 0) C both and ae simila to the cicle of adius with cente. Theefoe is simila to C 2. Edit this solution Solution: 2 Algebaic manipulations This solution woks with the equations of cicles and models the geometic tansfomations with algebaic manipulations. 4

5 Illustative (a, b) a. The cicle of adius centeed at is given by the solutions of the equation (x a ) 2 + (y b ) 2 = 2. To tansfom this into the equation fo the cicle of adius we may fist emove the fom x a and the b fom y b by substituting x = x a and y = y b. With these new coodinates ou equation becomes ( x 2 ) + ( ) =. This is the equation of a cicle of adius centeed at in - coodinates. We can x y x = x now scale and taking and and, with these coodinates, ou equation becomes the equation of a cicle of adius in - coodinates. Note, compaing this method to the fist solution, that moving fom - coodinates to - coodinates coesponds to tanslating the cicle by in the diection and in the -diection. The second substitution x = x and y = coesponds to the dilation by a facto of. y y 2 2 (0, 0) x y x x + y = x y x y x y a x b y a C C 2 b. Pat (a) applies to any cicle in the plane. If and ae two cicles then by pat (a) C C 2 (0, 0) C both and ae simila to the cicle of adius with cente. Theefoe is simila to C 2. G-C Simila cicles Typeset May 4, 206 at 2:00:45. Licensed by Illustative unde a Ceative Commons Attibution-NonCommecial-ShaeAlike 4.0 Intenational License. 5

4.2. Co-terminal and Related Angles. Investigate

4.2. Co-terminal and Related Angles. Investigate .2 Co-teminal and Related Angles Tigonometic atios can be used to model quantities such as