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1 Slide 1 / 199 Slide / 199 New Jersey enter for Teaching and Learning rogressive Mathematics Initiative This material is made freely available at and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others. Geometry Similar igures lick to go to website: Slide / 199 Table of ontents Slide / 199 click on the topic to go to that section atios and roportions Similar olygons using Transformations atios and roportions Similar olygons using orresponding arts eturn to the Table of ontents Similar Triangles roportions of Similar Triangles Similar ircles Solve roblems using Similarity Slide 5 / 199 Slide / 199 ratio - is a comparison between two quantities, in the same unit a and b, where b 0. efore learning about similar figures, we need to review ratios and proportions. ratio can be expressed three ways a to b, a:b, or a. b

2 Slide 7 / 199 Slide 8 / 199 xample xample Simplify the ratio. Simplify the ratio. The length of a rectangle is 9 inches. The width of a rectangle is feet. Write the ratio of the rectangle's width to length. 1 meters to 0 meters emember a ratio must be written in the same unit ow many inches in a foot? click click Slide 9 / 199 Simplify the ratio. 10 days to 5 weeks Simplify the ratio. 1 boys to 1 girls to 1 : 1/1 to :8 1 Slide 10 / 199 /7 :5 10/5 Slide 11 / 199 Slide 1 / 199 Simplify the ratio. proportion - is a statement that two ratios are equal. 00 feet to 1 mile (int: 1 mile = 5,80 feet) To solve a proportion, use the cross-product property. If a = c, then ad = bc b d

3 Slide 1 / 199 Slide 1 / 199 xample xample Solve for y. Solve for y. Use the cross-product property. K = y 9=y 8(y+) = 1y 8y+1 = 1y -8y -8y 1 = y =y Slide 15 / 199 K Use the cross-product property. Slide 1 / 199 Try this... Solve for y. Solve for x. 5 Slide 17 / 199 Slide 18 / 199 Solve for y Solve for x.

4 Slide 19 / 199 Slide 0 / 199 xample More roportion roperties Tell whether the statement is True or alse. If ab = cd, then b = d a c click If ab = cd, then a c click If b =d, then TU a c If b = d, then a+b = c+d click b d If, then simplify Slide 1 / 199 Slide / 199 xample xample Tell whether the statement is True or alse. Tell whether the statement is True or alse. If, then, then If Slide / 199 Slide / If Try this... omplete.. If, then. If, then True??? alse, then 1. If, then

5 Slide 5 / If, then True alse, then Slide 7 / If Slide 8 / 199, then James, Michelle and ngela have $50 in a ratio of :5:, respectively. ow much money do they each have? 8 If Slide / 199 James' amount + Michelle's amount + ngela's amount = $50 x + 5x + x = 50 10x = 50 x=5 James' amount = x = (5) = $10 Michelle's amount = 5x = 5(5) = $5 ngela's amount = x = (5) = $15 $10 + $5 + $15 = $50 Slide 9 / 199 x = 15 The distance between Trenton and Washington is 15 miles. Try this... Maria and Omar have $75 in a ratio of 9 to. ow much do they each have? The scale on a map of the ast oast US is 1 inch = 00 miles On the map, the distance between Trenton, NJ and Washington, is 0.7 inches. What is the actual distance between Trenton and Washington,? Slide 0 / 199

6 Slide 1 / 199 Slide / Students at the John. Kennedy Middle School built a 11-foot model of the Space Needle, using a scale of 1:55. What is the actual height of the Space Needle? 1 Three candidates in a recent election split the vote in a ratio of to 5 to. There were 0,000 votes cast. ow many votes did the winner receive? Slide / 199 0,000 0, ,000 10,000 Slide / The perimeter of a bedroom is 5 ft. The ratio of the length to the width is 5:. What is the width of the bedroom? Students type their answers here Similar olygons using Transformations eturn to the Table of ontents What does it mean for two figures to be similar? ongruent figures have exactly the same shape and size. When two figures are congruent you can translate (slide), reflect (flip) or rotate (turn) one so that it fits exactly on the other one. 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. an you identify any real life examples that use similar figures? Slide / 199 When two figures are congruent, you can map one figure onto the other by translating (sliding), reflecting (flipping), and rotating (turning). If two figures are similar, what transformations can you do to map one figure onto the other? Slide 5 / 199

7 Slide 7 / 199 Slide 8 / 199 eview Transformation Notation nd Vector Notation eview: Translation of to ''' was made by moving right 8 units and up units, we use the following notation: (x, y) (x+8, y+) transformation is a function that changes the position, shape, and/or size of a figure. The input is the pre-image. The output is the image. The translation vector used to translate to ''' is written as: '=<8,> Translations, reflections and rotations are rigid motions. rigid motion transformation changes the position of a figure. The shape and size are not changed. ilations do not change the shape of the figure. The size is changed. Therefore, dilations preserve angle measure. oordinate notation ilation (x, y) (x, y) Translation (x, y) (x, y-7) Slide 9 / 199 Slide 0 / 199 eview Notation eview Transformation Notation ' ' reflection over the x-axis uses the following notation: (x, y) (x, -y) ' ' ' ' ' ' 90 counter-clockwise 180 rotation about rotation about the the origin: origin: (x, y) (-x, -y) (x, y) (-y, x) Slide 1 / counterclockwise or 90 clockwise rotation about the origin: (x, y) (y, -x) Slide / 199 escribe the composition of similarity transformations needed to map to '''' to ''''''''. eview Notation ilation is mapped to ''' with center of dilation at the origin by: (x, y) (x, y) ~''''~'''''''' ' ' ' " " ' " " reflection over the y-axis uses the following notation: (x,y) (-x,y) '

8 Slide / 199 Slide / 199 oes the order in which you perform the similarity transformations matter? '''''''' by a dilation and a translation. ilation (x, y) (x, y) 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? '''' by a translation '''' '''''''' by a dilation ~''''~'''''''' ' ' ~''''~'''''''' ' 10 ' " ' ' " " -10 " ' ' " - - ' ' "-10 " " " ' 10 " " " Slide 5 / 199 Slide / Which similarity transformations can map ST to YZX? igures not drawn to scale S What is the translation vector ' used to translate Q to 'Q''? Q' ' 1 ' Q 5o T Y otation, dilation, translation X 5o Z ' '''', by a dilation '''' '''''''' by a translation Translation, dilation, translation eflection, dilation, translation ll of the above Slide 7 / 199 Slide 8 / If the scale factor of the dilation in the sequence of similarity transformations that map ST to YZX is and S= mm. ind the length of YZ. S None of the above 1 5o T Y X 5o igures not drawn to scale S Z 1/ o Y T X 15 ind the scale factor for the dilation that maps ST to YZX.? 5o igures not drawn to scale Z

9 Slide 50 / 199 Similar, the translation Similar, the dilation ' (x, y) (x+5,y)maps to. similarity transformations that can map to. ' Similar, the dilation ' (x, y) (1/x,1/y)maps to. Similar, the dilation (x, y) (1/x,1/y) and translation (x', y') (x'+5,y') map to. (x, y) (x,y) maps to '''. Similar, the dilation (x, y) (x,y) and translation (x', y') (x'+,y'+) map to '''. Slide 51 / 199 Slide 5 / Use the definition of similarity in terms of similarity transformations to determine if the two figures are similar. (x,-y) maps to. Not similar, there are no similarity transformations that can map to. Similar, the dilation (x, y) (x,y) maps to. Wrap Up: Similar, the reflection (x, y) Not similar, there are no Similar, the translation Using the definition of similarity in terms of similarity transformations, what do you have to do to show that two figures are similar? (x, y) (x+,y+) maps to '''. Not similar, there are no similarity transformations that can map to '''. 18 Use the definition of similarity in terms of similarity transformations to determine if the two figures are similar. 17 Use the definition of similarity in terms of similarity transformations to determine if the two figures are similar. Slide 9 / 199 Similar, the dilation (x, y) (x,y) and reflection (x', y') (x',-y') map to. Slide 5 / 199 Slide 5 / 199 Measurement olygon 1 olygon atio Similar olygons using orresponding arts eturn to the Table of ontents omplete the conjecture - If two figures are similar then corresponding angles are click congruent and the lengths of corresponding sides are proportional. click orresponding sides are proportional if the click ratios of their lengths are equal. click Teacher Notes When a figure is enlarged, how are corresponding angles related? ow are corresponding side lengths related? lick to complete Lab 1 - Similar olygons

10 Slide 55 / 199 Slide 5 / 199 When a figure is enlarged, how are corresponding angles related? ow are corresponding side lengths related? Similar olygons Lab Solutions click Measurement olygon 1 olygon When a figure is enlarged, how are corresponding angles related? ow are corresponding side lengths related? atio lick for interactive website to investigate. omplete the conjecture - If two figures are similar then corresponding angles are click congruent and the lengths of corresponding sides are proportional. click orresponding sides are proportional if the click ratios of their lengths are equal. click Slide 57 / 199 Slide 58 / If you start with a triangle with side lengths cm, 8 cm and 10 cm and angle measures a, b, and c. an I create a triangle that is similar if I dd to each side? The new triangle has sides 8 cm, 10 cm and 1 cm.. Subtract from each side? The new triangle has sides cm, cm, and 8 cm.. Multiply each side by? The new triangle has sides 1 cm, 1 cm, and 0 cm.. ivide each side by? The new triangle has sides cm, cm, and 5 cm. Teacher Notes lick to complete Lab - What does "lengths of corresponding sides are proportional" mean? has side lengths 7, 9 and 11. Which triangle has corresponding side lengths that are proportional? 10, 1, 1 What does "lengths of corresponding sides are proportional" mean?,, 8 1, 7, ll of the above What conclusions can you make about side lengths and similar triangles? What operations provide triangles that are similar to the original? Slide 59 / 199 XYZ has side lengths 18, 5, and 0. Which triangle has corresponding side lengths that are proportional?, 15, 0 15,, 57 ll of the above 1, 8, Sketch two nonsimilar polygons whose corresponding angles are congruent. 1 Slide 0 / 199

11 Slide 1 / 199 Slide / 199 Sketch two nonsimilar polygons whose corresponding sides are proportional but whose corresponding angles are not congruent. '''' 1. List all pairs of congruent angles ' ' ' '. Write the ratio of corresponding sides in a statement of When or more proportionality ratios are equal, you can write an extended proportion. Slide / 199 The ratio of the lengths of corresponding sides is the similarity ratio. an you write another similarity statement? '''' ' ' ' ' ' Y ST ~ XYZ ST ~ YXZ ' ST ~ YZX ' None of the above ' ecide whether the polygons are similar. If so, write a similarity statement. '''' ' Slide / 199 What is the relationship between the scale factor k and the similarity ratio r for ~ ''''? ' Slide 5 / 199. ind the similarity ratio r from to '''', reduced to lowest terms. ' ' '''' S 1 5o X 5o T igures not drawn to scale Z 1. ind the similarity ratio r from '''' to, reduced to lowest terms. Slide / 199

12 Slide 7 / 199 Slide 8 / 199 Write another similarity statement for List all pairs of congruent angles for the similarity statement ST ~ YZX. ST~ YZX Y ST~ ZYX 5o X 5o Z T igures not drawn to scale ll of the above None of the above None of the above Slide 9 / 199 Slide 70 / Write the statement of proportionality for the similarity statement ST ~ YZX. ind the similarity ratio from ST to YZX. Y 1 5o Slide 71 / 199 5o X Z T igures not drawn to scale Slide 7 / ~G.5 7o 1 o igures not drawn to scale o G o None of the above Q 5o S.7 10o 5. L 7. M 8 10o ~G ind the similarity ratio from SQ to LMNO 5o 7 ecide whether the polygons are similar. If so, write a similarity statement. None of the above 1 None of the above ~G S TS~ XZY O. igures not drawn to scale N S

13 Slide 7 / 199 Slide 7 / 199 M No 7o 1o. 1o 5 o 5 1 o. 7o L 5 o 7.5 N Yes 1o 0 Given ST~ UVW, a student explained that and are a pair of corresponding sides and and are a pair of corresponding sides. The student wrote the proportion. Is this correct? If not, rewrite the proportion. 9 ecide whether the figures are similar. Yes igures not drawn to scale No Slide 75 / 199 Slide 7 / 199 o the properties of congruence apply to similarity? xplain why congruence is a special case of similarity. eflexive roperty of ongruent Triangles very triangle is congruent to itself Slide 77 / 199 Transitive roperty of ongruent Triangles o the properties of congruence apply to similarity? Symmetric roperties of ongruent Triangles Slide 78 / 199 o the properties of congruence apply to similarity? K J L K J L

14 Slide 79 / 199 Slide 80 / 199 G NQLM. Solve for the variables. G NQLM. Solve for the variables. o w 5y 9o 9o N w 5y G G Slide 81 / 199 Slide 8 / 199 Try this... QS KJML. Solve for the variables. 5 xo z J Q M wo 11 o K. 9 L 1 S 1 0 o x+ z o y G w z o y x+ 87o o 1 x+ G What is the value of y? (~G) 87o w Slide 8 / 199 What is the value of w? (~G) Slide 8 / w 87o y 10 1 What is the value of x? (~G) 1 o z o y Q o N 5 Q 1 5 int: orresponding angles are congruent xo int: orresponding angles are congruent xo 1 M z L M z L xample G

15 Slide 85 / 199 Slide 8 / 199 Wrap Up: w 87o x+ 1 o z o G y Using the definition of similarity in terms of corresponding parts, what do you have to do to show that two figures are similar? What is the value of z? (~G) Slide 87 / 199 Slide 88 / 199 To show that two figures are similar: Show that all corresponding angles are congruent and corresponding sides are proportional. Similar Triangles Show that there is a sequence of similarity transformations that map one figure to the other. The shortcuts for triangles are:,sss, SS eturn to the Table of ontents In this unit, you will learn shortcuts to prove that two triangles are similar. Slide 89 / 199 Slide 90 / 199 #1 ngle-ngle Similarity ( ~ ) ostulate If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. To prove this, click for Lab - ngle ngle Similarity ( ~ ) Measurement Triangle Triangle ''' atio ' ' If < # < and < # <, then '. Therefore, If two angles of one triangle are congruent to two angles click of another triangle, then the triangles are similar. Teacher Notes

16 Slide 91 / 199 Slide 9 / 199 xample Let's also prove this using similarity transformations. To prove the triangles are similar, find a sequence of similarity transformations that map to XYZ. etermine whether the triangles are similar. If they are similar write a similarity statement. If they are not similar, explain why. easons Given rove: Z ~ XYZ ilate Z with scale factor k = Z ~ ''Z K L efinition of scale factor efinition of dilation M corresponding angles of ~ triangles are congruent Statements Given: N simplify Transitive roperty of ''Z XYZ S ''Z~ XYZ efinition of Z ~ XYZ Transitive roperty of ~ Slide 9 / 199 Slide 9 / 199 Try this... ecide whether the triangles are similar. If they are similar, write a similarity statement. If they are not similar, explain why. xample etermine whether the triangles are similar. If they are similar, write a similarity statement. If they are not similar, explain why. 8o T X 55o Z Slide 9 / 199 re the triangles similar? 8o Yes No G 58o Slide 95 / re the triangles similar? o 58 o Y Yes No o 1o 8o S 55o

17 Slide 97 / 199 Slide 98 / 199 To prove this, click for Lab # - Side-Side-Side Similarity (SSS ~ ). 7 ' If Measurement Triangle Triangle ''' atio ' ' 1, then the triangles are similar. Teacher Notes # Side-Side-Side Similarity (SSS ) Theorem If the corresponding sides of two triangles are proportional, then the triangles are similar. 7cm 1cm 9cm 18cm 11cm cm Therefore, If the corresponding sides of two triangles are proportional, then the triangles are similar. click Slide 99 / 199 Slide 100 / 199 Let's also prove this using similarity transformations. To prove the triangles are similar, find a sequence of similarity transformations that map Z to XYZ. Statements Given: rove: Let's prove the triangles are similar another way. easons hoose on rove: Given Z ~ XYZ ilate Z with scale factor k = Z ~ ''Z Statements Given: ~ orresponding angles ostulate Y simplify XYZ efinition of Z ~ XYZ Transitive roperty of ~ XYZ ~ orresponding sides of ~ triangles are proportional simplify ''Z~ YQ~ Q simplify SSS uler ostulate arallel ostulate eflexive roperty of efinition of dilation XYZ so that =Y raw Z efinition of scale factor ''Z XYZ easons X Given If =Y then =YQ and =Q roperty of = YQ ~ YQ ~ XYZ Slide 101 / 199 SSS roperty of Transitive roperty of ~ Slide 10 / 199 xample etermine whether the triangles are similar.if they are similar write a similarity statement. If they are not similar, explain why. xample etermine whether the triangles are similar.if they are similar write a similarity statement. If they are not similar, explain why K To identify corresponding sides, list the sides from smallest to greatest. 1 L Write the statement of proportionality.. S.8

18 Slide 10 / 199 Slide 10 / 199 Let's prove the triangles are similar another way. # Side-ngle-Side Similarity (SS ) Theorem If two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are similar. Statements Given: hoose on rove: ~ arallel ostulate orresponding angles ostulate eflexive roperty of XYZ Z YQ~ ~ XYZ orresponding sides of ~ triangles are proportional Q and < <, then the If uler ostulate raw easons so that =Y 's are similar. Y Given X roperty of = If =Y then =YQ Given YQ SS ~ YQ roperty of ~ XYZ Slide 105 / 199 Transitive roperty of ~ Slide 10 / 199 Let's also prove this using similarity transformations. To prove the triangles are similar, find a sequence of similarity transformations that map to XYZ. To prove this, click for Lab #5 - Side-ngle-Side Similarity (SS ~ ) Given: Statements easons Given Measurement Triangle Triangle ''' atio ' 18 ' 1 rove: Z ~ XYZ ' Z ~ cm 1cm 9cm 18cm ilate Z with scale factor k = ''Z efinition of scale factor efinition of dilation corresponding angles of ~ triangles are congruent simplify simplify Transitive roperty of Therefore, If two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are similar. click Slide 107 / 199 O efinition of Z ~ XYZ Transitive roperty of ~ SS 5 5 G 0 0 M 10 XYZ 15 ''Z~ Try this... etermine whether the triangles are similar.if they are similar write a similarity statement. If they are not similar, explain why. N XYZ Slide 108 / 199 xample etermine whether the triangles are similar.if they are similar write a similarity statement. If they are not similar, explain why. L ''Z

19 Slide 109 / 199 Slide 110 / 199 Yes 5 T Yes 1 1 U No V 9 5 S Slide 111 / 199 Slide 11 / re the triangles similar? 0 Which is not a method to prove triangles are congruent? Yes SSS SS No L Slide 11 / 199 Slide 11 / 199 xample Given G ind G. 1 Which is not a method to prove triangles are similar? SSS L. SS W 9 7 x G No 8 re the triangles similar? 7 re the triangles similar?

20 Slide 115 / 199 Slide 11 / 199 The triangles are similar therefore K ~ K N M 7 K 5o Z J L 1 alse 5 x True Z. ind Z. 8 Slide 117 / K ~ Z. ind K. K 7 o.8 15 Z x 5o Z. ind m. K 5o K ~ 8 o o 7o 5o.8 Z 7 o 15 Slide 118 / Slide 119 / 199 Slide 10 / 199 Wrap Up: ll equilateral triangles are similar. ow do you prove that two triangles are similar? alse True Is ngle-ngle-side (S) a shortcut to prove that two triangles are similar?

21 Slide 11 / 199 Slide 1 / 199 o you remember this question from the last section? ecide whether the triangles are similar.if they are similar write a similarity statement. If they are not similar, explain why. o 58 G 58o roportions of Similar Triangles eturn to the Table of ontents Write the ratio of corresponding sides in a statement of proportionality. click Slide 1 / 199 Since, what can we say about and Slide 1 / 199? Side Splitter Theorem If a line parallel to one side of a triangle intersects the other two sides, then it divides the sides proportionally. by the onverse of orresponding ngles ostulate If two lines are cut by a transversal and the corresponding angles are congruent, then the lines are parallel. Using the statement of proportionality show that. o 58 If, then. G 58o Slide 15 / 199 Slide 1 / 199 rove the Side Splitter Theorem. Given: easons Given orresponding angles ostulate eflexive roperty of G~ rove: xample Statements If a line parallel to one of a triangle intersects the other two sides, then it divides the sides proportionally. ~ orresponding sides of ~ triangles are proportional += G+G= G Segment ddition ostulate Substitution Simplify Simplify Subtraction roperty of = roperty of proportions J ind JK. I 1 K 9 L

22 Slide 17 / 199 Slide 18 / 199 xample ind x. Try this... ind x. I 10 1 x J L 1 x K 1 8 Slide 19 / 199 Slide 10 / 199 rove the onverse to the Side Splitter Theorem. onverse to the Side Splitter Theorem If a line divides two sides of a triangle proportionally, then the line is parallel to the third side. Statements If a line divides two sides of a triangle proportionally, then the line is parallel to the third side. Given: rove: easons Given ddition roperty of = Simplify += G+G= Segment ddition ostulate Substitution If, then G eflexive roperty of G~ Slide 11 / 199 SS~ orresponding angles of ~ triangles are congruent orresponding angles onverse Slide 1 / 199 xample xample an you prove if an you prove if?? I 8 7 N 9 J

23 Slide 1 / 199 Slide 1 / Is //? Yes Q S 5 T 1 10 No 5 Try this... an you prove if Q // ST? Slide 15 / 199 Slide 1 / Is? 9 ind y. No y Yes Slide 17 / 199 Slide 18 / ind y. 51 ind y. y 15 y 1 1

24 Slide 19 / 199 Slide 10 / 199 onstructing Similar Triangles using the onverse to the Side Splitter Theorem Step xtend of the sides of the triangle by doubling the lengths. Step 1 raw a triangle. Use a ruler to ensure that at least sides of the triangle are whole numbers. inches inches Slide 1 / 199 Step onnect the extended sides. Label the new segment. y SS, the small triangle is similar to the large triangle. inches inches inches.5 inches inches Slide 1 / 199 Triangle ngle isector Theorem If a ray bisects an angle of a triangle and intersects the opposite side of the triangle, then the ray divides the opposite sides into segments that are proportional to the other two sides of the original triangle. Slide 1 / 199 rove the Triangle ngle isector Theorem. If a ray bisects an angle of a triangle and intersects the opposite side of the triangle, then the ray divides the opposite sides into segments that are proportional to the other two sides of the original triangle. Given: inches inches inches.5 inches inches bisects inches Slide 11 / 199 If.5 inches inches inches.5 inches inches, bisects rove: Statements efinition of intersect Side Splitter Theorem orresponding ngles ostulate Given 1 then easons arallel ostulate = efinition of angle bisector lternate Interior ngles Theorem Transitive roperty of ongruence Transitive roperty of ongruence onverse of the ase ngles Theorem Substitution

25 Slide 15 / 199 Slide 1 / 199 xample xample ind x. ind K. G x x K 15-x Slide 17 / 199 Slide 18 / 199 Try this... ind the length of O. 5 omplete the proportion. I 5-y Slide 19 / 199 J K Slide 150 / ind y. 5 ind the length of K. I J 1 y 10 K 8 I y 10 J 0-y 5 K y O 15 5 G

26 Slide 151 / 199 Slide 15 / ecide whether ST. 0 1-x K I J Yes 18 x No 1 iscuss the difference between dividing segments proportionally and dividing segments equally. 5-x T x Slide 155 / 199 U T Wrap Up: S V 15 Slide 15 / ind the length of T. 0 S Slide 15 / Q ind the length of JK. Slide 15 / 199 re all circles similar? ow can you prove this? Similar ircles Show that there is a set of similarity transformations that map one figure to the other. eturn to the Table of ontents Show that all corresponding angles are congruent and corresponding sides are proportional. owever, circles do not have angles or sides. What can you do? ircles have a radius, diameter and circumference.

27 Slide 157 / 199 Slide 158 / 199 escribe the composition of similarity transformations needed to map circle to circle ' to circle ". (Note: is the center of the dilation) circle ~ circle ' ~ circle " circle with radius ~ circle with radius ' ~ circle " with radius "" escribe the composition of similarity transformations needed to map circle to circle ' to circle ". (Note: the origin is the center of the dilation) ' ' " " " Slide 159 / 199 Slide 10 / 199 oes the order in which you perform the similarity transformations matter? The order matters ONLY I the center of the circle is the origin. The center of the circle will move TWI in each case, but in different directions. circle circle ' circle ' by a translation circle '' by a dilation b. If the translation is done first, followed by a dilation, the center of the circle follows the translation directions, and then the new center moves out on a line from the origin. circle circle ' If the center of the circle is point, the center only moves ON, for the translation. The center stays in the same place for the dilation. That is why it does not matter whether the translation or dilation is done first. circle ' by a dilation circle " by a translation Slide 11 / 199 Slide 1 / Which similarity transformations can map circle to circle "? Origin is the center of the dilation. circle circle '' by a translation and a dilation. Translation (x, y) (x+, y-) ilation (x', y') (x', y') What is the scale factor k of the dilation? " ' ' " ilation (x, y) (0.5x, 0.5y) and translation (x', y') (x'+,y'+). (x, y) (x+,y+) and Translation dilation (x', y') (0.5x', 0.5y'). ll of the above None of the above " ' a. If the dilation is done first, followed by a translation, the center of the circle moves out on a line from the origin, and then follows the translation directions. oes the order in which you perform the similarity transformations matter?

28 Slide 1 / 199 Slide 1 / 199 translation "=<1,> " " ilation with scale factor of 1/ translation "=<1,> ilation with scale factor of, translation "=<,1> ilation with scale factor of 1/, ' translation "=<,1> 0 Which similarity transformations can map circle with center (-1,) and radius to circle with center (,) and radius 5? oint is the center of the dilation. ilation circle to circle ' by scale factor of 5/ and translation (x,y) (x+,y+) Translation (x, y) (x+,y+) and dilation of circle to circle ' by a scale factor of 5/. ll of the above None of the above Slide 15 / 199 dilation of cirle to circle ' by a scale factor of 1/. Translation (x, y) (x,y-8) and dilation of circle to circle ' by a scale factor of. ind the scale factor for the dilation that maps the small circle to the large circle. 1 Which similarity transformations can map circle with center (0,) and radius to circle with center (0,-) and radius? oint is the center of the dilation. Translation (x, y) (x,y+8) and Slide 1 / 199 ilation circle to circle ' by a scale factor of and translation (x, y) (x, y-8). ilation circle to circle ' by a scale factor of 1/ and translation (x, y) (x, y-8). Slide 17 / 199 Slide 18 / 199 ind the scale factor for the dilation that maps circle to circle '..5 ind the scale factor for the dilation that maps the large circle to the small circle. ilation with scale factor of, circle) to circle "? oint is the center of dilation. 59 Which similarity transformations can map circle (large annot be determined

29 Slide 19 / 199 Slide 170 / 199 hoose the vector that describes the translation. " If the scale factor of the dilation in the sequence of similarity transformations that map circle to circle is and the radius of circle is 5, find the radius of circle. Slide 171 / 199 Slide 17 / 199 When a circle is enlarged or reduced, what changes? radius, diameter, circumference rove all circles are similar. Given: or all circles the ratio of the circumference to the diameter is always the same, regardless of the dimensions of the circle. circle with radius r circle with radius s rove: circle is similar to circle That ratio is always equal to or, more easily, what we call. s Since this ratio never changes, we can say that all circles are similar. Statements easons definition of translation definition of translation the center of circle ' is definition of translation definition of dilation r definition of dilation transitive property of ~ Slide 17 / 199 Slide 17 / 199 Wrap Up: xplain how to use a reflection and a dilation to prove circle is similar to circle '. " Solve roblems using Similarity eturn to the Table of ontents

30 Slide 175 / 199 Slide 17 / 199 ow can you use similar figures to solve real-life problems? Grand anyon National ark, Z Slide 177 / 199 irst, construct right triangle. 1. Identify a landmark at point.. lace a marker at point directly across from point.. Walk to point, place a marker and measure the distance of. Then, construct right triangle. 1. Walk to point, place a marker and measure the distance of.. Walk to point, place a marker and measure the distance of. ow can you prove that ~? ow can you find the distance across the Grand anyon? Slide 179 / 199 ow can we find the height of the Washington Monument? Washington Monument and eflecting ool, Washington.. Slide 178 / 199 Why? Why? ~ Why? ow do you find d? Write a statement of proportionality that uses d. The distance across the Grand anyon is 00 ft. Slide 180 / 199 We are going to use shadows to find the height of the Washington Monument. This is another method of indirect measurement. What is the difference between direct measurement and indirect measurement? Using similar triangles and indirect measurement, you will find large distances and the heights of trees, flagpoles, and buildings. ow can we find the distance across the Grand anyon?

31 Slide 181 / The right triangle formed by the Washington Monument and its shadow.. The right triangle formed by you and your shadow. Measure the lengths of the shadows. ow can you prove that ~? ow can you find the height of the Washington Monument? Slide 18 / 199 ow can we find the height of the Washington Monument when there are no shadows? Why? Why? ~ Why? ow do you find h? Write a statement of proportionality that uses h. On a sunny day, the sun's rays cast a shadow on a vertical object. There are two similar right triangles. Slide 18 / 199 The height of the Washington Monument is 555 ft. Slide 18 / 199 We are going to use a mirror trick to find the height of the Washington Monument. This is another method of indirect measurement. lace a mirror with cross hairs (an X) drawn on it flat on the ground between yourself and the Washington Monument. Look into the mirror and walk to a point at which you see the top of the Washington Monument lining up with the mirror's cross hairs. The light rays from the top of the Washington Monument to the mirror and back up to your eye form equal angles. Slide 185 / 199 In hysics, Slide 18 / 199 Measure the distance from you to the mirror and the Washington Monument to the mirror. ow can you prove that ~? ow can you find the height of the Washington Monument?

32 Slide 187 / 199 Slide 188 / 199 Why? 7 lamppost casts a 9 ft shadow at the same time a person ft tall casts a ft shadow. ind the height of the lamppost. Why? ~ Why? ow do you find h?.7 ft 1.5 ft 15 ft The height of the Washington Monument is 555 ft. Slide 189 / 199 Slide 190 / To find the width of a river, you use a surveying technique as shown. Setup the proportion to find the distance across the river. g 1 ft You 5 ft in ft 15 ft 18.9 ft Slide 191 / 199 Solve a problem using similar rectangles. ectangle is 8 inches by 11 inches ectangle G is x inches by inches There is a margin of 1 inch along all of the edges. ind x so that ~ G If x =. inches, then ~G 5 ft 8 Your little sister wants to know the height of the giraffe. You place a mirror on the ground and stand where you can see the top of the giraffe as shown. ow tall is the giraffe? 189 in ft Write a statement of proportionality that uses h. Slide 19 / 199 Or ectangle is 8 inches by 11 inches ectangle G is inches by x inches There is a margin of 1 inch along all of the edges. ind x so that ~ G If x = 8.5 inches, then ~G

33 Slide 19 / 199 Slide 19 / 199 typographic grid system is a set of horizontal and vertical lines that determine the placement of type on a page. The lines create rows and columns of identical rectangles. The length has inches = - (for the margin) and requires 5 rows of rectangles graphic designer wants to design a new grid system for a poster. The rectangle needs to be similar to the poster. The poster is inches by inches. The grid must have margins of 1 inch along all edges. There must be 5 rows of rectangles. The rectangles must be similar in size to the poster. The width has inches = (-) ow many columns of rectangles will fit? 1. What should be the length x of the small rectangle?. What should be the width y of the small rectangle?. ow many columns of small rectangles can there be? Should there be or 5 columns of rectangles? If there are 5 columns of rectangles, you will not have a 1 inch border. Therefore, columns of rectangles. Slide 195 / 199 Slide 19 / 199 What is the other solution? 70 graphic designer wants to design a new grid system for a poster. The poster is 5 cm by 7 cm. The grid must have margins of cm along all edges. There must be 5 rows of rectangles. The rectangles must be similar in size to the poster. The length has inches = - (for the margin) and requires 5 rows of rectangles The rectangle needs to be similar to the poster. What should be the length x of the rectangles? The width has inches = (-) ow many columns of rectangles will fit? 10. cm 5 columns of rectangles. 1. cm 1. cm Slide 197 / 199 Slide 198 / cm 7 graphic designer wants to design a new grid system for a poster. The poster is 5 cm by 7 cm. The grid must have margins of cm along all edges. There must be 5 rows of rectangles. The rectangles must be similar in size to the poster. ow many columns of rectangles can there be? 1. cm 5 1. cm 10 cm 71 graphic designer wants to design a new grid system for a poster. The poster is 5 cm by 7 cm. The grid must have margins of cm along all edges. There must be 5 rows of rectangles. The rectangles must be similar in size to the poster. What should be the width y of the rectangles? 10 cm

34 Slide 199 / 199 Wrap Up: xplain the similarities and differences between direct measurement and indirect measurement.