Geometry Progressive Mathematics Initiative This material is made - PDF document

Slide 1 / 199 Slide 2 / 199 New Jersey Center for Teaching and Learning Geometry Progressive Mathematics Initiative This material is made freely available at www.njctl.org and is intended for the non-commercial use of Similar Figures students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning 2014-02-05 community, and/or provide access to course materials to parents, students and others. www.njctl.org Click to go to website: www.njctl.org Slide 3 / 199 Slide 4 / 199 Table of Contents click on the topic to go to that section Ratios and Proportions Ratios and Proportions Similar Polygons using Transformations Similar Polygons using Corresponding Parts Return to the Similar Triangles Table of Contents Proportions of Similar Triangles Similar Circles Solve Problems using Similarity Slide 5 / 199 Slide 6 / 199 ratio - is a comparison between two quantities, in the same unit a and b, where b ≠ 0. Before learning about similar figures, we need to review ratios and proportions. A ratio can be expressed three ways a a to b, a:b, or . b

Slide 7 / 199 Slide 8 / 199 Example Example Simplify the ratio. Simplify the ratio. The length of a rectangle is 9 inches. The width of a rectangle is 2 16 meters to 40 meters feet. Write the ratio of the rectangle's width to length. Remember a ratio must be written in the same unit How many inches in a foot? click click Slide 9 / 199 Slide 10 / 199 2 Simplify the ratio. 10 days to 5 weeks 1 Simplify the ratio. 12 boys to 16 girls A 2 to 1 A 4:3 Answer Answer B 2/7 12/16 B 3 to 4 C C 2:5 D 6:8 D 10/35 Slide 11 / 199 Slide 12 / 199 3 Simplify the ratio. proportion - is a statement that two ratios are equal. 300 feet to 1 mile (Hint: 1 mile = 5,280 feet) Answer To solve a proportion, use the cross-product property. a c If , then ad = bc = b d

Slide 13 / 199 Slide 14 / 199 Example Example Solve for y. Solve for y. Use the cross-product property. Use the cross-product property. CHECK 8(y+2) = 12y 8y+16 = 12y -8y -8y 36 = 4y 16 = 4y 9 = y CHECK 4 = y Slide 15 / 199 Slide 16 / 199 Try this... 4 Solve for x. Solve for y. A 3 Answer 4 B Answer 5 C 6 D Slide 17 / 199 Slide 18 / 199 5 Solve for x. 6 Solve for y. A 3 A 3 Answer Answer 4 B B 4 5 C 5 C 6 D D 6

Slide 19 / 199 Slide 20 / 199 Example More Proportion Properties Tell whether the statement is True or False. a c b d If , then = = b d a c click If , then a b a c If , then = = b d c d click TRUE a c a+b c+d If , then = b d = If , then b d click simplify Slide 21 / 199 Slide 22 / 199 Example Example Tell whether the statement is True or False. Tell whether the statement is True or False. If , then If , then Answer Answer Slide 23 / 199 Slide 24 / 199 7 If , then Try this... Complete. True 1. If , then ? Answer Answer False 2. If , then ? ? 3. If , then

Slide 25 / 199 Slide 26 / 199 8 If , then 9 If , then True A B C Answer Answer False Slide 27 / 199 Slide 28 / 199 10 If , then James, Michelle and Angela have $50 in a ratio of 2:5:3, respectively. How much money do they each have? A B C James' amount + Michelle's amount + Angela's amount = $50 Answer 2x + 5x + 3x = 50 10x = 50 x = 5 James' amount = 2x = 2(5) = $10 Michelle's amount = 5x = 5(5) = $25 Angela's amount = 3x = 3(5) = $15 $10 + $25 + $15 = $50 Slide 29 / 199 Slide 30 / 199 The scale on a map of the East Coast US is Try this... 1 inch = 200 miles On the map, the distance between Trenton, NJ and Maria and Omar have $75 in a ratio of 9 to 6. How much do Washington, DC is 0.76 inches. What is the actual distance they each have? between Trenton and Washington, DC? Answer x = 152 The distance between Trenton and Washington DC is 152 miles.

Slide 31 / 199 Slide 32 / 199 11 Students at the John F. Kennedy Middle School 12 Three candidates in a recent election split the vote built a 11-foot model of the Space Needle, using a in a ratio of 2 to 5 to 6. There were 260,000 votes scale of 1:55. What is the actual height of the cast. How many votes did the winner receive? Space Needle? A 20,000 B 40,000 Answer Answer 100,000 C 120,000 D Slide 33 / 199 Slide 34 / 199 13 The perimeter of a bedroom is 54 ft. The ratio of the length to the width is 5:4. What is the width of the bedroom? Students type their answers here Similar Polygons using Transformations Answer Return to the Table of Contents Slide 35 / 199 Slide 36 / 199 What does it mean for two figures to be similar? When two figures are congruent, you can map one figure onto the other by translating (sliding), reflecting (flipping), and Congruent figures have exactly the rotating (turning). same shape and size. When two figures are congruent you can translate (slide), If two figures are similar, what transformations can you do to reflect (flip) or rotate (turn) one so that map one figure onto the other? it fits exactly on the other one. Answer Similar figures have the same shape but may NOT be the same size. The turtle on the right is an enlargement of the turtle on the left. These turtles are similar figures. Can you identify any real life examples that use similar figures?

Slide 37 / 199 Slide 38 / 199 Review Transformation Notation And Vector Notation Review: Translation of ABC to A'B'C' A transformation is a function that changes the position, was made by moving right 8 units shape, and/or size of a figure. and up 3 units, we use the following notation: (x, y) (x+8, y+3) The input is the pre-image. The output is the image. The translation vector used to Translations, reflections and rotations are rigid motions. translate ABC to A'B'C' is A rigid motion transformation changes the position of a written as: AA'=<8,3> figure. The shape and size are not changed. Dilations do not change the shape of the figure. The size is changed. Therefore, dilations preserve angle measure. Coordinate notation - Dilation (x, y) (2x, 2y) Translation (x, y) (x, y-7) Slide 39 / 199 Slide 40 / 199 Review Notation Review Transformation Notation A' A' B' B C B C C B C' C' C' B' A A A' B' A 270 counter- clockwise or 90 counter-clockwise 180 rotation about A reflection over the x-axis 90 clockwise A reflection over the y-axis uses rotation about the the origin: uses the following notation: rotation about the following notation: origin: (x, y) (-x, -y) the origin: (x, y) (x, -y) (x,y) (-x,y) (x, y) (-y, x) (x, y) (y, -x) Slide 41 / 199 Slide 42 / 199 Describe the composition of similarity transformations needed to Review Notation map ABCD to A'B'C'D' to A''B''C''D''. Dilation ABCD~A'B'C'D'~A''B''C''D'' ABC is mapped to A'B'C' C' Answer B' with center of dilation at the origin by: C B (x, y) (2x, 2y) D' D A' A B" C" D" A"

Slide 43 / 199 Slide 44 / 199 Does the order in which you perform the ABCD A''B''C''D'' by a dilation and a translation. similarity transformations matter? Dilation (x, y) (2x, 2y) Translation (x', y') (x', y'-7) Yes. In this case, if you first perform the translation and then the dilation, the image is not the same. What is the scale factor k of the dilation? Answer ABCD A'B'C'D', by a dilation ABCD A'B'C'D' by a translation A'B'C'D' A''B''C''D'' by a translation A'B'C'D' A''B''C''D'' by a dilation ABCD~A'B'C'D'~A''B''C''D'' C' B' ABCD~A'B'C'D'~A''B''C''D'' C C' B B' 10 D' B C D A' A B" C" C B D' 2 A' A D D -10 A -2 2 10 B" C" -2 C' D' B' A' D" C" A" D" -10 D" A" B" A" Slide 45 / 199 Slide 46 / 199 What is the translation vector PP' used to 14 Which similarity transformations can map translate PQR to P'Q'R'? RST to YZX? Figures not drawn to scale S 1 3 52 o Answer R P' Q' 2 T Y Answer 4 2 52 o A Rotation, dilation, translation X Z 6 R' P Q B Translation, dilation, translation R C Reflection, dilation, translation D All of the above Slide 47 / 199 Slide 48 / 199 15 Find the scale factor for the dilation that maps 16 If the scale factor of the dilation in the sequence of similarity transformations that map RST to YZX is 3 RST to YZX. and RS=6 mm. Find the length of YZ. S A 1/2 1 3 S A 2 6 52 o R Answer B 2 2 T Y Answer 52 o 4 2 R Y B 3 T 52 o ? C 4 X Z 6 52 o X Z C 9 Figures not drawn to scale D None of the above Figures not drawn to scale D 18