Notes 1.2. Notes 1.3
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1 Notes 1.2 Comparing Similar Figures * image: A. Complete the instructions for Stretching a Figure on page 8 using Labsheet 1.2. Tell how the original figure and the image are alike and how are they different. Compare these features: Shapes Lengths of line segments and perimeters Areas Angles * What do you think would have happened if we had used a three-band stretcher? * corresponding sides: * corresponding angles: Notes 1.3 Corresponding Parts Label all corresponding parts of the following similar triangles: Sides Angles
2 Notes 2.1 Drawing Wumps A. Everybody wants to be like Mug! Remember to always start with Mug Wump s coordinates as you follow the rules to fill out the table with each of the other characters coordinates. B. Draw Mug, Zug, Lug, Bug and Glug on separate coordinate graphs. For parts 1-3 of each figure, plot the points in order. Connect them as you go along. (Part 1 will make the body, Part 2 is the mouth, and Part 3 is the nose.) For part 4, plot the two points but do not connect them. (These are the eyes.) C. Which wumps belong to Mug s family? Who are the imposters? D. Describe the similarities and differences of each character with Mug. What happened to each character compared to Mug? Why? Zug: Lug: Bug: Glug: * Rules for similar figures: * Rules for imposters:
3 Notes 2.3 Scale Factors 1 Mug 4 The Wump Mouths Above are the mouths of Mug, Zug, Lug, Bug and Glug. Study their dimensions to determine which mouth belongs to which character and fill in the table below. Wump Width of Nose Length of Nose Width Length (ratio) Perimeter (2L + 2W) Area (L x W) Mug (x, y) Zug (2x, 2y) Bug (3x, 3y) Hug (4x, 4y) Snug (5x, 5y) (10x, 10y) (20x, 20y) (100x, 100y) Nug (nx, ny) Lug (3x, y) Glug (x, 3y) What patterns and relationships do you notice in the table between similar wumps? Compare their rules, side lengths, ratios, perimeters and areas. 2.3
4 * similar: * scale factor: The diagram on page 26 shows the mouths (rectangles) and noses (triangles) from the Wump family and from the imposters. A. After studying the noses and mouths in the diagram, Marta and Zack agree that rectangles J (Zug) and L (Mug) are similar. However, Marta says the scale factor is 2, while Zack says it is 0.5. Is either of them correct? How would you describe the scale factor so there is no confusion? B. Decide which pairs of rectangles are similar and find the scale factor each way. (Enlarging from small big and shrinking from big small) C. Decide which pairs of triangles are similar and find the scale factor each way. (Enlarging from small big and shrinking from big small) D. For each pair of similar figures, how can you use the scale factor to predict the relationship between: 1. Perimeters: 2. Areas: E. Explain how to find the scale factor from a figure to a similar figure.
5 Notes 3.3 Scale Factors and Similar Shapes A. For parts 1-3, draw a rectangle similar to rectangle A that fits the given description. Find the base and height of each new rectangle. Also, find the enlarging and shrinking scale factors between rectangle A and the new rectangle. 1. The scale factor from rectangle A to the new rectangle is 2.5. Base: Height: Enlarging S.F. Shrinking S.F. 2. The area of the new rectangle is ¼ the area of rectangle A. Base: Height: Enlarging S.F. Shrinking S.F. 3. The perimeter of the new rectangle is 3 times the perimeter of rectangle A. Base: Height: Enlarging S.F. Shrinking S.F. B. For parts 1-2, draw a triangle similar to triangle B that fits the given description. Find the base and height of each new triangle. Also, find the enlarging and shrinking scale factors between triangle B and the new triangle. 1. The area of the new triangle is 9 times the area of triangle B. Base: Height: Enlarging S.F. Shrinking S.F. 2. The scale factor from triangle B to the new triangle is ½. Base: Height: Enlarging S.F. Shrinking S.F.
6 3.3 *Hints for Part C: Match the corresponding parts (long side with long side, etc.) All angles in a triangle add up to 180. C. Rectangles ABCD and EFGH are similar. 1. Find the length of side AD. Show your work. Triangles ABC and DEF are similar. 1. By what number do you multiply the length of side AB to get the length of side DE? 2. Find the missing side lengths and angle measures. Show your work.
7 Notes 4.1 Ratios Within Similar Parallelograms * ratio: * equivalent ratios: A. The lengths of two sides are given for each rectangle. 1. For each rectangle, find the ratio of the length of a short side to the length of a long side. Write each ratio as a fraction and decimal. A: B: C: D: 2. What do you notice about the ratios for similar rectangles? About the ratios for non-similar rectangles? 3. For each pair of similar rectangles, find the scale factor from the smaller rectangle to the larger rectangle and from the larger rectangle to the smaller. (These are your enlarging and shrinking scale factors!) and and and small large: small large: small large: large small: large small: large small: What information does the scale factor give about two similar figures? 4. Compare the information given by the scale factor to the information given by the ratios of side lengths.
8 4.1 B. For each parallelogram, find the ratio of the length of a longer side to the length of a shorter side. E: F: G: 1. How do the ratios compare? 2. Which of the parallelograms are similar? Explain. C. If the ratio of adjacent side lengths in one parallelogram is equal to the ratio of the corresponding side lengths in another, can you say that the parallelograms are similar? Explain. * How to show shapes are similar:
9 Notes 4.3 Finding Missing Parts For questions A C, each pair of figures is similar. Find the missing side lengths. Explain or show how you got your answer. A. x = B. x = C. x =
10 D. The figures are similar. Find the missing measurements and explain or show how you got your answers. a = b = c = d = e = f = x = E. The figures in your book (p.65) are similar. The measurements are shown in inches. 1. Find the value of x: 2. Find the value of y: 3. Find the scale factor enlarging and shrinking the figure. Small big: Big small: 4. Find the area of one of the figures: (big / small) How could you use the scale factor to find the area of the other figure? Which scale factor would you use? Explain and find the area of the other figure.
11 Notes 4.3 Extra Practice Show your work for each question! A. Find the missing measurements of the similar figures below. What is the scale factor from the larger shape to the smaller shape above? B. The three rectangles are similar. Find their missing measurements. C. The triangles are similar, but not drawn exactly to scale. Find the missing side lengths, and answer the questions below. 1. What is the scale factor from the big triangle to the little triangle? 2. What is the scale factor from the little triangle to the big triangle? 3. How many times greater is perimeter of the big triangle compared to the little triangle? 4. How many times greater is the area of the big triangle compared to the little triangle?
12 Notes 5.1 Using Shadows Suppose you want to use the shadow method to estimate the height of a building. You make the following measurements: length of the stick: 3 m length of the stick s shadow: 1.5 m length of the building s shadow: 8 m A. Make a sketch of the building, the stick, and the shadows. Label each given measurement. Label the height of the building x or? B. Use similar triangles to find the building s height from the given measurements. Show your work. Building height: C. A tree casts a 25-foot shadow. At the same time, a 6-foot stick casts a shadow 4.5 feet long. Sketch a picture and label all the known measurements. Use similar triangles and show your work to find out: How tall is the tree? D. A radio tower casts a 120-foot shadow. At the same time, a 12-foot high basketball backboard (with pole) casts a shadow 18 feet long. Sketch a picture and label all the known measurements. Use similar triangles and show your work to find out: How high is the radio tower?
13 Notes 5.2 Using Mirrors A. Jim and Su use the mirror method to estimate the height of a traffic signal near their school. They make the following measurements: height from the ground to Jim s eyes: 150 cm distance from the middle of the mirror to Jim s feet: 100 cm distance from the middle of the mirror to a point directly under the traffic signal: 450 cm 1. Make a sketch. Show the similar triangles formed and label the given measurements. 2. Show your work to find the height of the traffic signal: B. Jim and Su also use the mirror method to estimate the height of the gymnasium in their school. They make the following measurements: Height from the ground to Su s eyes: 130 cm Distance from the middle of the mirror to Su s feet: 100 cm Distance from the middle of the mirror to the gym wall: 9.5 m *Notice the difference in units! You must convert all the measurements to be in the same units in order to find the correct answer!! 1. Make a sketch. Show the similar triangles formed and label the given measurements. 2. Show your work to find the height of the gymnasium:
14 Notes 5.3 Finding Lengths Within Similar Triangles A. Use the river diagram above. Which triangles appear to be similar? Explain. B. Draw the similar triangles separately. Label the measurements that you know. Then use these similar triangles to find the distance across the river from Stake 1 to Tree 1. Show your work! C. Identify, draw, and label the similar triangles separately from the diagrams below. (Diagrams are not drawn to scale) Find the missing side length x. Show your work and circle your final answer! 1. 2.
15 Review 1. How do you find the enlarging and shrinking scale factors between two similar shapes? 2. Find the missing sides. Show your work! (Pictures are not drawn to scale) 3. A tower casts a shadow 144 feet long. Mrs. Way is 5 10 and her shadow is 12 feet long. Sketch a picture and show your work to find the height of the tower.
16 4. Amy is standing 8 feet away from a mirror. The distance of her eyes to the ground is 5 6. The mirror is 4 feet away from a St. Bernard dog. Sketch a picture and show your work to find the height of the dog. 5. Find x. Show your work.
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