Geometry Unit 3 Similar Figures and Dilations Name Unit 3 Similar Figures and Dilations
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1 Name 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 Name 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 Ato B. 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 Name 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) In the diagram, FGHJ ~LMNP. 1. What is the scale factor of LMNP to FGHJ? x Find the value of x. 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, similarity statement TSU~LKM 2 2. NO 3. YES, linear scale factor = 2, similarity statement RSTU~WXYZ 1 4. a) Scale factor = 5 b) x = 27.5, y = 12, z = x = x = 9 7. XY = 8 8. YZ = 8 9. C and D 10. A, D and E 11. Answers may vary 12. Answers may vary 13. x = 4 so DE = x = 15 so EP =
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: 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 ABCand its image. b) The vertices of ΔABC is A(-3, 4), B(3.5, -5), C(2, 3). Find the vertices of the dilated image with scale factor of 2. The center of the dilation is the origin. Example 3: 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? Unit 3.2 Worksheet Answers 1. Answers may vary (check with classmates or teacher for verification) 2. Answers may vary (k > 1) 3. Q (-40.5, 45) 4. LSF = LSF = LSF = 3 ; x = 10; Enlargement 2 7. A ( 1.5, 3 ),Y ( 2, 2.5 ), B ( 3.5, 0.5 ) D ( 4, 0.5 )Sum = A ( 0,6 ),B ( 4, 6 ), C ( 8, 8 ) Product = A ( 1, 2 ),B ( 3, 1 ), C ( 2, 1 )Sum = Area = 96 in k = A ( 1.5, 1.5 ),B ( 1, 1 ), C (3.5, 1.5 )
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: Identify corresponding parts Given the similar trapezoids ABCD and EHGF below, identify the side that is proportional tobc. Example 2:Find lengths of unkown corresponding parts The two rectangular swimming pools are similar. How far is the diagonall across the smaller pool? 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! 25 meters The scale factor is Example 3: Use similarity to find lengths A high school wants to build a basketball court that is similar to an NBA basketball court, which is 94 feet long and 50 feet wide. Unforunately, the high school can only budget a room for a court that is 30.8 feet longer than it s width. How long should the court be, to the nearest foot? x ft x+30.8 ft Example 4: 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 2) YOU TRY NOW! a) Given the similar trapezoids ABCD and EFGH below, identify the side that is proportional to BC. b) A 42.9 ft flagpole casts a ft long shadow. About how long is the shadow of a 6.2 ft tall woman? c) Parallelograms ABCD and EFGH are similar. What is the length ofgh? ABCD~JKLM LK = QUESTIONS OR REFLECTION Write down at most 2 questions that you can ask the next day. BE SPECIFIC. 1)
8 Annotate Here
9 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
10 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:
11 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 x = 1 Transformation #2: Dilate by a factor of 1 centered at the 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 the origin Rotation: 90 clockwise about the origin C' D' C" D" CCW K (3, 2)
12 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 18. 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) 19. C 3,0 D 2, 6, C 1,0 D ( 2 3, 2) 20. C 6, 12 D 4,0, C 8, 12 D (6,0) 21. P 3,6 Q 9,0 R (6, 3), P 6,3 Q (0,9)R (3,6) 22. P 2,6 Q 6,2 R (4,0), P 1,3 Q (3,1)R (2,0) 23. 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 = SUM = SUM = SUM = SUM = 3
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