Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Möbius transformation wikipedia , lookup
Scale invariance wikipedia , lookup
Trigonometric functions wikipedia , lookup
History of trigonometry wikipedia , lookup
Euler angles wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Euclidean geometry wikipedia , lookup
Unit 5: Similarity Lesson 3: Similar Figures In our study of transformations, we have seen many figures that remain congruent after translations, reflections, and rotations [these were called rigid transformations]. We have also seen figures which retain their shape but do not remain the same size after applying a size transformation (also called a dilation). These figures and their images after a size transformation are similar figures. Definition: Two figures are similar if one is the image of the other under a size transformation of scale factor, k, or under a composite of transformations (one of which much be a size change). That is to say, one figure is a dilation of the other. Similar figures are figures that are the same shape, but not necessarily the same size. If these figures should also be the same size, the figures are called congruent. Similar triangles - scale factor 2 Are the figures below similar figures? 1. Are the figures the same shape? ____________ 2. Do they appear to have the same angles? ____________ 3. What do you observe is different between the two triangles? In order to be similar, we need to be able to perform transformations (translations, rotations, reflections, and dilations) on one triangle to create the other. 4. What are the coordinates of ΔABC? A _______ B_______ C_______ 5. Reflect ΔABC over the y-axis. What are the coordinates of the reflected points? A' _______ B'_______ C'_______ 6. Now plot the points for the reflected triangle A'B'C'. 7. Now focus on ΔA'B'C' and ΔLMN. Name pairs of corresponding angles. ______ & ______ ______ & ______ ______ & ______ 8. Comparing the coordinates of the corresponding angles, what appears to be the scale factor applied to ΔA'B'C'? We now have similar triangles since we could apply transformations that mapped ΔABC onto ΔNML 9. Fill in the blanks to summarize the above: Starting with ΔABC, we first _________ over the ________. Then we _____________ by a scale factor of _______ . Now, writing this as a similarity transformation rule: (x, y) → ( ______ , _____ ) 10. You are told to perform a similarity transformation that is the composition of the following two transformations (performed in the order given) a. Describe the following transformations: Transformation I: (x, y)→(23x, 23y) Transformation II: (x, y)→(x,− y) A size change of scale factor ________ A reflection over the _______________. b. ΔPQR has vertices P(6, 12), Q(18, 0), R(12, −6). Using the transformations from part (a), what would be the vertices of the final image? P (6, 12) Q (18, 0) R (12, -6) P' ________ Q' ________ R' ________ P'' ________ Q'' ________ R'' ________ c. Write a rule that combines the two transformations from above into a similarity transformation. (x, y) → ( ______ , ______ ) Two polygons with the same number of sides are SIMILAR provided that 1) Their corresponding angles have the same measure and 2) The ratios of lengths of corresponding sides is a constant (this constant is the s cale factor). In the above diagram, quadrilateral A'B'C'D' ~ quadrilateral A BCD. The symbol ~ means “is similar to.” Proof that they are similar: 1) Corresponding angles have the same measure: m∠ A'= m∠ A m∠ B'= m∠ B m∠ C'= m∠ C m∠ D'= m∠ D 2) Ratios of lengths of corresponding sides is a constant: The constant 5 2 is called the scale factor from quadrilateral ABCD to quadrilateral A 'B'C'D'. It scales (multiplies) the length of each side of quadrilateral ABCD to produce the length of the corresponding side of quadrilateral A'B'C'D'. Questions: 1. If the scale factor from A BCD to A'B'C'D' is 5/2, what is the scale factor from A'B'C'D' to ABCD? 2. If two figures are similar, how can we find the scale factor from the smaller figure to the larger figure? 3. Follow up question: how do we find the scale factor from the larger figure to the smaller figure? 4. Suppose ΔPQR ~ ΔXYZ and the scale factor from ΔPQR to ΔXYZ is 34 . What is true about the corresponding angles? What is true about pairs of corresponding sides? Be specific and list corresponding angles and sides in your answer below. State if the polygons are similar by finding the ratios of corresponding sides. The polygons in each pair are similar. Find the missing side length. The polygons in each pair are similar. Solve for x. The polygons in each pair are similar. Find the missing side length. The polygons in each pair are similar. Solve for x. Knowing that two triangles are similar allows you to conclude that the three pairs of corresponding angles are congruent, and that the three pairs of corresponding sides are related by the same scale factor. Conversely, if you know that the three pairs of corresponding angles are congruent and the three pairs of corresponding sides are related by some scale factor, you can conclude that the triangles are similar! 33. Each triangle described in the table below is similar to ΔABC. For each triangle (ΔDEF, ΔGHI, ΔJKL), use this fact and the additional information given to fill in the table. Triangle Angle Measures ΔABC m∠ A = 64° m∠ B = 18° m∠ C = 98° ΔDEF m∠ D = ____° m∠ E = 64° m∠ F = 18° Triangle Angle Measures ΔAB C ΔGH I m∠ A = 64° m∠ B = 18° m∠ C = 98° m∠ G = ____° m∠ H = ____° m∠I = ____° Triangle Angle Measures ΔAB C ΔJK L Shortest Side Length Longest Side Length Third Side Length AC = 4.0 AB = 12.8 BC = 11.6 Scale Factor from ΔABC 2 Shortest Side Length AC = 4.0 Longest Side Length AB = 12.8 IG = 6.4 Shortest Side Length m∠ A = 64° m∠ B = 18° m∠ C = 98° AC = 4.0 m∠ J = ____° m∠ K = 18° m∠ L = 98° JL = 14.0 Longest Side Length AB = 12.8 Third Side Length Scale Factor from ΔABC BC = 11.6 GH = 5.8 Third Side Length BC = 11.6 Scale Factor from ΔABC