Foundations of Math II Unit 3: Similarity and Congruence Academics High School Mathematics
3.1 Warm Up 1. Jill and Bill are doing some exercises. Jayne Funda, their instructor, gently implores Touch your nose to your knees, maggots! Their attempts to please Ms. Funda are shown below. Bills says, I m doing better than you, Jill. My nose is much closer to my knees! Jill replies, That isn t a fair comparison, Bill. With whom do you agree? Who is doing a better job? Explain your answer. Jill Bill 2. The perimeter of ΔCOW is 12 units. a) Find possible lengths for CO, OW, and CW. b) Find four more sets of possible lengths. c) How many answers are possible? Adapted from Geometry: A Moving Experience developed by the Curriculum Research & Development Group, College of Education at the University of Hawaii 1
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3.2 Warm Up 1. Which of the figures below could be the image of figure a when dilated? Explain why or why not for each figure. p g a c e r f s 2) a) Draw a line that passes through the origin of a coordinate plane and forms a 45 angle with the x-axis. b) Find the coordinates of at least three points on the line. c) Write an equation for the line. What do you notice? Adapted from Geometry: A Moving Experience developed by the Curriculum Research & Development Group, College of Education at the University of Hawaii 6
3.2 Practice with Dilations on the Coordinate Plane Graph three points that lie in three different quadrants and connect them to form a triangle. Label the vertices of the triangle as TRI. Record the coordinates of the triangle in the table below. Then find and apply the algebraic rules for each of the scale factors listed below. Graph and label each image. Scale Factor 3 2 2 1 2 Algebraic Rule (x, y) (x, y) (x, y) T (, ) T (, ) T (, ) T (, ) R (, ) R (, ) R (, ) R (, ) I (, ) I (, ) I (, ) I (, ) What would each scale factor be if written as a percent? 7
Explain why or why not for each pair. 8
Find the scale factor. The pre-image is indicated by an arrow. 9
3.3 Warm Up 1. Draw each of the following dilations of quadrilateral BRIA: a. 150% scale factor using center X. b.!! scale factor using center Y. c. 1.5 scale factor using center I. d. What do you notice? A B X Y I R Adapted from Geometry: A Moving Experience developed by the Curriculum Research & Development Group, College of Education at the University of Hawaii 10
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3.4 Warm Up 1. ΔRAP is an image of ΔCON using a dilation. Find point Z, the center of dilation, and also the scale factor. C O A R N P Adapted from Geometry: A Moving Experience developed by the Curriculum Research & Development Group, College of Education at the University of Hawaii 18
3.4 Investigation: Geometric Mean 1. Use your protractor to find the measure of the following angles. m ABC = m BCD = m CAB = m BDC = m ADC = m BCD= m DCA= What type of segment is CD called in ABC? What kind of triangles are ΔABC, ΔADC, and ΔBDC? 1. Trace ABC with a ruler on patty paper and carefully cut out ADC and BDC. You will need to place the letter of each vertex in the interior of each triangle on the patty paper so that you can still tell what you are working with after you cut it out (see below). 2. Next trace and cut out ABC. Be sure to label each vertex in the interior of the triangle. 3. Stack the cut out triangles so that that corresponding sides match up. Note that the three right triangles are similar to each other. Write a similarity statement for the three triangles. ΔABC 19
We know that the ratios of corresponding sides of similar triangles are proportional,!" 4. Using BDC and ΔCDA fill in the proportion: =!" So DC must be the geometric mean of and. 5. Using BDC and ABC fill in the proportion:!"!" = So BC must be the geometric mean of and. 6. Using ADC and ABC, fill in the proportion:!"!" = So AC must be the geometric mean of and. a h b m n c 7. Write three proportions for the picture above using what you learned from this activity. 20
Geometric Mean: Example Problems 1. 2. 3. 4. 21
3.4 Geometric Mean Practice 22
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3.5 Warm Up 1) a) If a line has a slope greater than 1, what angle might it make with the x-axis? b) If a line has a slope less than 1, what angle might it make with the x-axis? c) If a line has a slope equal to 1, what angle might it make with the x-axis? Adapted from Geometry: A Moving Experience developed by the Curriculum Research & Development Group, College of Education at the University of Hawaii 24
3.5 Midsegment Example Problems Example 1 Find x. Example 2 DE is the midsegment of ΔABC. Find x, AC, and ED. 7x 28 Example 3 MN is the midsegment of ΔJKL. MN = 2x + 1 KJ = 5x 8 Find x, MN, and KJ. Example 4 25
3.5 Midsegments Show What You Know! 1) XY is the midsegment of ΔRST. Find each requested measure based on the given information. a) XY = 16, RS =? b) RS = 22, XY =? c) XY = 5x, RS = 15, x =? d) m R = 23, m TXY =? e) m XYS = 137, m YSR 2) Find x and y. 3) Find MS, PT, and ST. 3y 18 4) a) b) c) 26
3.6 Warm Up 1) a) A line forms an angle measuring less than 45 with the x-axis. What might its slope be? b) A line forms an angle measuring more than 45, but less than 90, with the x-axis. What might its slope be? c) What might the slope be if the line forms an obtuse angle with the x-axis? Adapted from Geometry: A Moving Experience developed by the Curriculum Research & Development Group, College of Education at the University of Hawaii 27
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3.7 Warm Up 1) A line passes through the origin and the point A(7, 3). Without graphing the line, what can you conclude about the angle it will form with the x-axis? Adapted from Geometry: A Moving Experience developed by the Curriculum Research & Development Group, College of Education at the University of Hawaii 30
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3.9 Flow Proof Examples 1) C is the midpoint of AD A D 1 2 AC DC ΔABC 2) MN PT 3 4 NO TO 1 2 ΔMNO 3) BD bisects ABC BD BD ABD CBD AB CB ΔABD 39
4) E is the midpoint of AD A D B C AE DE ΔABE 5) AS bisects MP Reflexive Property of Congruence Given ΔMAS 6) Given Definition of Angle Bisector ΔABD 40
3.10 Warm- up 1. Erica builds a ramp that makes a 45 with the ground. Her support board is 10 feet from the beginning of the ramp. a. How high is her support board? b. How long is her ramp? 2. Line m forms a 40 angle with the x- axis. Find the slope of line m. Explain your answer? Adapted from Geometry: A Moving Experience developed by the Curriculum Research & Development Group, College of Education at the University of Hawaii 41
3.10 Practice 1) C is the midpoint of BE C is the midpoint of AD ΔABC ΔDEC 2) 1 and 2 are right angles AD CB 3) All right angles are congruent ΔABD ΔCDB AB A S S Converse of the Isosceles Triangle Theorem ΔABD ΔCBD 42
3.11 Warm Up 1) Bill builds a ramp at a 56 angle with the ground. He uses a 12-foot support board and finds that the support board must be 8 feet from the beginning of the ramp in order to make the 56 angle. Jill also builds a ramp at a 56 angle with the ground. She uses a 9-foot support board. How far should her board be from the beginning of her ramp? Illustrate and explain your answer. Adapted from Geometry: A Moving Experience developed by the Curriculum Research & Development Group, College of Education at the University of Hawaii 43
Vocabulary Word Definition Characteristics Picture and/or Symbol Real Life Examples AAS ASA congruent dilation 44
Vocabulary Word Definition Characteristics Picture and/or Symbol Real Life Examples extremes flow proof geometric mean means 45
Vocabulary Word Definition Characteristics Picture and/or Symbol Real Life Examples midsegment Midsegment Theorem proof proportion 46
Vocabulary Word Definition Characteristics Picture and/or Symbol Real Life Examples SAS scale factor side similar 47
Vocabulary Word Definition Characteristics Picture and/or Symbol Real Life Examples SSS triangle Triangle Angle Sum Theorem vertex 48
Vocabulary Word Definition Characteristics Picture and/or Symbol Real Life Examples 49
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