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Similarity, Right Triangles, and Trigonometry

Similarity, Right Triangles, and Trigonometry. Clusters: 1. Understand similarity in terms of similarity transformations 2. Prove theorems involving similarity. 3. Define trigonometric ratios and solve problems involving right triangles. 4. Apply trigonometry to general triangles.

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Similarity, Right Triangles, and Trigonometry

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  1. Similarity, Right Triangles, and Trigonometry Clusters: 1. Understand similarity in terms of similarity transformations 2. Prove theorems involving similarity. 3. Define trigonometric ratios and solve problems involving right triangles. 4. Apply trigonometry to general triangles.

  2. Learning Target • I can define dilation. • I can perform a dilation with a given center and scale factor on a figure in the coordinate plane.

  3. Connection to previews lesson… • Previously, we studied rigid transformations, in which the image and preimage of a figure are congruent. In this lesson, you will study a type of nonrigid transformation called a dilation, in which the image and preimage of a figure are similar.

  4. Dilations A dilation is a type of transformation that enlarges or reduces a figure but the shape stays the same. The dilation is described by a scale factor and a center of dilation.

  5. Dilations The scale factor k is the ratio of the length of any side in the image to the length of its corresponding side in the preimage. It describes how much the figure is enlarged or reduced.

  6. Reduction: k = = = Enlargement: k = = CP CP CP CP 5 2 1 2 3 6 PQR ~ P´Q´R´, is equal to the scale factor of the dilation. P´Q´ PQ The dilation is a reduction if k < 1 and it is an enlargement if k > 1. P P´ 6 5 P´ P 2 3 Q Q´ R • • C C R´ Q R´ Q´ R

  7. Constructing a Dilation Examples of constructed a dilation of a triangle.

  8. Steps in constructing a dilation Step 1: Construct ABC on a coordinate plane with A(3, 6), B(7, 6), and C(7, 3).

  9. Steps in constructing a dilation Step 2: Draw rays from the origin O through A, B, and C. O is the center of dilation.

  10. Steps in constructing a dilation Step 3: With your compass, measure the distance OA. In other words, put the point of the compass on O and your pencil on A. Transfer this distance twice along OA so that you find point A’ such that OA’ = 3(OA). That is, put your point on A and make a mark on OA. Finally, put your point on the new mark and make one last mark on OA. This is A’.

  11. Steps in constructing a dilation Step 3:

  12. Steps in constructing a dilation Step 4: Repeat Step 3 with points B and C. That is, use your compass to find points B’ and C’ such that OB’ = 3(OB) and OC’ = 3(OC).

  13. Steps in constructing a dilation Step 4:

  14. Steps in constructing a dilation You have now located three points, A’, B’, and C’, that are each 3 times as far from point O as the original three points of the triangle. Step 5:Draw triangle A’B’C’. A’B’C’ is the image of ABC under a dilation with center O and a scale factor of 3. Are these images similar?

  15. Steps in constructing a dilation Step 5:

  16. Questions/ Observations: Step 6: What are the lengths of AB and A’B’? BC and B’C’? What is the scale factor? • AB = 4 • A’B’= 12 • BC = 3 • B’C’= 9

  17. Questions/ Observations: Step 7: Measure the coordinates of A’, B’, and C’. Image A´(9, 18) B´(21, 18) C´(21, 9)

  18. Questions/ Observations: Step 8: How do they compare to the original coordinates? P(x, y) P´(kx, ky) Pre-image A(3, 6)  B(7, 6)  C(7, 3)  Image A´(9, 18) B´(21, 18) C´(21, 9)

  19. Draw a dilation of rectangle ABCD with A(2, 2), B(6, 2), C(6, 4), and D(2, 4). Use the origin as the center and use a scale factor of . How does the perimeter of the preimage compare to the perimeter of the image? 1 2 C´ 1 1 O y x A´ In a coordinate plane, dilations whose centers are the origin have the property that the image of P(x, y) is P´(kx, ky). SOLUTION Because the center of the dilation is the origin, you can find the image of each vertex by multiplying its coordinates by the scale factor. D C A B D´ A(2, 2)  A´(1, 1) B(6, 2)  B´(3, 1) B´ • C(6, 4)  C´(3, 2) D(2, 4)  D´(1, 2)

  20. Draw a dilation of rectangle ABCD with A(2, 2), B(6, 2), C(6, 4), and D(2, 4). Use the origin as the center and use a scale factor of . How does the perimeter of the preimage compare to the perimeter of the image? 1 2 C´ 1 O x 1 y A´ In a coordinate plane, dilations whose centers are the origin have the property that the image of P(x, y) is P´(kx, ky). SOLUTION From the graph, you can see that the preimage has a perimeter of 12 and the image has a perimeter of 6. D C A B D´ A preimage and its image after a dilation are similar figures. B´ • Therefore, the ratio of the perimeters of a preimage and its image is equal to the scale factor of the dilation.

  21. Example 3 Determine if ABCD and A’B’C’D’ are similar figures. If so, identify the scale factor of the dilation that maps ABCD onto A’B’C’D’ as well as the center of dilation. Is this a reduction or an enlargement?

  22. Assignment/Homework Work with a partner in the classwork on “Constructing Dilation” Homework: Answer Guided Practice page 510 #12 to 15.

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