Unit 3SimilarFigures and Dilations
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1 Unit 3 Similar Figures and Dilations Unit 3SimilarFigures and Dilations Target 1 Use proportions to identify lengths of corresponding parts in similar figures Target 2 Perform and identify dilations Target 3 Use ratios of lengths, perimeter, & area to determine unknown corresponding parts 3.3a Use Scale Factor & Similarity to Determine Unknown Lengths in Polygons & Circles 3.3b Use Scale Factor & Similarity to Determine Unknown Corresponding Parts Target 4 Perform compositions of figures to determine the coordinates and location of the image Name:
2 Unit 3 Similar Figures and Dilations 3.1 Similar Figures Target 1 Use proportions to identify lengths of corresponding parts in similar figures Vocabulary Similar Polygons: Corresponding angles are and corresponding sides are Annotate Here Linear Scale Factor: between Example 1: Use similarity statements In the diagram, ΔABC~ΔDEF. 1. List all pairs of corresponding sides 2. Check that the ratios of corresponding side lengths are equal. Ratio 1: Ratio 2: Ratio 3: Linear Scale Factor of ΔABC to ΔDEF? Example 2: Determine triangle similarity A) Determine whether the polygons are similar. If they are, write the similarity statement and find the scale factor of AtoB. Similar? YES or NO Linear Scale Factor: Similarity Statement: B) Which of the following triangle measurements represents a similar triangle to one with measurements of 6, 8, and 10 inches? (a) 18 in, 24 in, and 30 in (b) 1.5 in, 2 in, and 2.5 in (c) 12 in, 16 in, and 18 in
3 Unit 3 Similar Figures and Dilations Example 3: Find linear scale factors and unknown side lengths Proportion - A comparison of a) ΔABC~ΔDEF. What is the scale factor of ΔABC to ΔDEF? What is the length of ED? b) Given that STU~ FED and ST = x + 2, UT = x 2 x 14, FE = 1, and DE = 2,find ST. Example 4: Use ratios to find an unknown side The lengths of the sides of a triangle have the ratio 1:2:3. If the perimeter of the triangle is 60 yards, what is the length of the smallest side? Unit 3.1 Worksheet Answers 1. YES, linear scale factor = 9,Answers vary, example similarity statement TSU~LKM 2 2. NO 3. YES, linear scale factor = 2,Answers vary, example similarity statement RSTU~WXYZ 1 4. a. Scale factor = 5 2 b. x = 27.5, y = 12, z = x = x = 9 7. XY = YZ = C and D 10. A, D and E 11. Answers may vary 12. Answers may vary 13. x = 4 so ST = 8 or x = 8 so ST = x = 10 so EP = 9 (-25 is extraneous) 15. 6
4 3.2 Dilations Target 2 Perform and identify dilations Vocabulary Dilation-A transformation in which a polygon is or by a given linearscale factorusing a centered. Annotate Here Linear Scale Factor Look at distance from to = Prime Distance PreImage Distance If k < 1, then image is a If k > 1, then image is an Example 1: Identify dilations Determine whether the dilation is a reduction (shrink) or an enlargement (expand). Find the linear scale factor of the dilation. a) b) c) d) 11
5 Example 2: Finding the Center of Dilation Example 3: Perform Dilations a) The vertices of triangle ABC are A (-3, 0), B (0, 6), C (3, 6). Use scalar multiplication to find A B C after a dilation with is center at the origin and a scale factor of 1 3. Graph ABC and its image. b) The vertices of ΔABC is A(-3, 4), B(3, -2), C(2, 3). Find the vertices of the dilated image with scale factor of 2. The center of the dilation is (0, 1). Example 4: Understanding Notation ΔABCis dilated to form triangle ΔA B C. If AB B C = 7, what is? A B BC QUESTIONS OR REFLECTION What concepts were important to take away from this target? Questions?
6 3.3 Day1 Find Unknown Lengths in Polygons&Circles Target 3 Use ratios of lengths, perimeter, &area to determine unknown corresponding parts Example 1:Find lengths of unkown corresponding parts The two rectangular swimming pools are similar. How far is the diagonall across the smaller pool? 25 meters Annotate Here FUN FACT! All circles are similar!all angles are congruent because circles have a360 angle. All lengths are proportional because radii and circumferences are proportional! Example 2: Use similarity to find lengths ABCD~JKLM 2x 2 x 6 1 3x + 2 The scale factor is... LK = Example 3: Use similarity to find lengths A 42.9 ft flagpole casts a ft long shadow. About how long is the shadow of a 6.2 ft tall woman?
7 YOU TRY NOW! a) The vertex of J of a regular hexagon has the coordinates (-6, 27). If the hexagon is dilated by a factor of 1. Note: Regular means that all sides of 5 a polygon are the same length and all interior angles are congruent. Annotate Here b) Given the similar trapezoids ABCD and EFGH below, identify the side that is proportional to BC. c) The vertex of B of an octagon is located at (24, -16). The octagon is dilated by a factor of 0.25, with the center of dilation at the origin. What are the coordinates of B? d) Parallelograms ABCD and EFGH are similar. What is the length ofgh? QUESTIONS OR REFLECTION Write down at most 2 questions that you can ask the next day. BE SPECIFIC. 1) 2)
8 3.3 Day 3 Find Unknown Perimeters and Areas Target 3 Use ratios of lengths, perimeter, & area to determine unknown corresponding parts A 6 m B 18 m What is the linear scale factor of Rectangle A to Rectangle B? What is the perimeter of Rectangle A? Rectangle B? What is the ratio of the perimeters of Rectangle A to Rectangle B? What is the area of Rectangle A? Rectangle B? What is the ratio of the areas of Rectangle A to Rectangle B? Perimeters of Similar Polygons If two polygons are similar with the lengths of corresponding sides in the ratio a:b, then the ratio of their perimeters is :. Linear Scale Factor: Side Length of Polygon 1 Side Length of Polygon 2 = Ratio of Perimeters: Linear Scale Factor/Ratio of the Perimeters Ratio of the Areas Example 1: Find an unknown ratio Linear Scale Factor Perimeter Ratio Area Ratio
9 Example 2: Find the perimeter of similar figures a) ΔWXY~ΔPQR. Find the perimeter of WXY. Linear Scale Factor Ratio of the Areas Perimeter of WXY: b) The ratio of the areas of two squares is 8:50. If the perimeter of the smaller square is 25 m, what is the perimeter of the larger square? Linear Scale Factor Ratio of the Areas Example 3: Find the areas of similar figures a) The ratio of the area of two circles is 9:16. If the area of the larger circle is 68 ft 2, what is the area of the smaller circle? Perimeter of Larger Square: Linear Scale Factor Ratio of the Areas Area of XYZ: b)δabc~δdef. AB = 3 inches, DE = 6 inches, and the area of ΔABC is 72 square inches. What is the area of ΔDEF? Linear Scale Factor Ratio of the Areas Area of DEF:
10 3.4 Compositions with Dilations Target 4 Perform compositions of figures to determine the coordinates and location of the image Example 1: Perform the composition a) Transformation #1: Reflect over the line y = x Transformation #2: Dilate by a scale factor of 1 centered at origin 2 Annotate Here Graph y = 2 Graph x = -1 Coordinates after each transformation ABC A B C A B C A( -7,1) A (, ) A (, ) B(-3, 4) C(-2,1) B (, ) C (, ) B (,) C (, ) Rotate K in a counterclockwise direction b) The endpoints of CD are C (-2, 3) and D (0, -2). Graph the image of CD after the composition. Dilate by a scale factor of 3 centered at (-1, 0) Rotation: 90 clockwise about (-1, 1) C' D' C" D" CCW K (3, 2)
11 Example 2:Describe the composition Describe the composition of transformations. Give the exact translation, reflection or rotation using proper notation. Transformation 1: Transformation 2: SUMMARY In your own words, describe what a composition is. 3.4 Worksheet Answers 16. a. P 9,6, P ( 3,2) b. P 8,16, P ( 8,16) c. P 1,2, P ( 1 2, 1) d. P 2, 1, P ( 2,1) 17. C 3,0 D 2, 6, C 1,0 D ( 2 3, 2) 18. C 6, 12 D 4,0, C 8, 12 D (6,0) 19. P 3,6 Q 9,0 R (6, 3), P 6,3 Q (0,9)R (3,6) 20. P 2,6 Q 6,2 R (4,0), P 1,3 Q (3,1)R (2,0) 21. Transformation 1: Dilate by a factor of 2 centered at the origin Transformation 2: Reflect over line x = Transformation 1: Rotate 90 CW about the origin Transformation 2: Dilate by a factor of 1 centered at the origin A 3.4,3.4, A 3.4,0.6, Sum = A 2,8, A 1,6,Sum = A 6, 5, A 3, 2.5, Sum = A 4, 2, A 2, 6, Sum = A 5, 9, A 1, 3, Sum = A 2,6, A 4,3, Sum = 7
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