BLoCK 4 ~ ratios, rates And PerCents

Transcription

1 BLoCK 4 ~ ratios, rates And PerCents circles and similarity Lesson 20 ParTs of circles Explore! What Is It? Lesson 21 circumference of a circle Explore! Round and Round Lesson 22 PeriMeTer and circumference Explore! Flower Garden Lesson 23 similar and congruent FigUres Explore! Match the Shapes Lesson 24 ratios and similar FigUres Explore! Ratio of Lengths and Perimeters review BLock 4 ~ circles and similarity corresponding PArts word wall center chord similar Figures circumference radius circle diameter central Angle π (Pi) congruent Figures 112 Block 4 ~ Ratios, Rates And Percents ~ Circles And Similarity

2 BLoCK 4 ~ CIrCLes And similarity tic - tac - toe haiku Write one haiku poem to explain each part of a circle. GaMe Board Create circular game boards with parts of circles shaded to win. VocABulAry QuiZ Create a study guide and quiz with vocabulary words from this textbook. See page 141 for details. See page 140 for details. See page 141 for details. Arc length Find the length of arcs on a circle using ratios and circumference. circumference Find the diameter and circumference of different balls and circular objects around your house. π Explore π using a spreadsheet. π See page 123 for details. See page 128 for details. See page 124 for details. mural Pictures missing Angles Sketch a mural with four combined shapes. Find the perimeter of the shapes. Find and label three sets of congruent shapes and three sets of similar shapes that are not congruent. Find missing central angles in circles. See page 128 for details. See page 132 for details. See page 118 for details. Block 4 ~ Circles And Similarity ~ Tic - Tac - Toe 113

3 Parts OF circles Lesson 20 explore! what is it? step 1: Draw a point on a piece of paper. This is a center point. a. Use a ruler to draw a point 3 inches from the center point. b. Use a ruler to draw at least 8 more points 3 inches from the center point. Be sure to include points to the left, right, above and below the center point. c. Connect the 9 points around the center point with a smooth curve. d. What shape have you made? e. If you measured from any point on the edge of the circle to the center point, what would the distance be between the points? step 2: Draw a new center point on your paper. a. Use a ruler to draw a point 2 centimeters from the center point. b. Use a ruler to draw at least 8 more points 2 centimeters from the center point. Be sure to include points to the left, right, above and below the center point. c. Connect the 9 points around the center point with a smooth curve. d. What shape have you made? e. If you measured from any point on the edge of the circle to the center point, what would the distance be between the points? step 3: You have created two different circles. How can you tell them apart? You created two circles of different sizes. Each circle consists of points that are equal distances from the center point. A circle is the set of all points the same distance from a point called the center. Th e radius of a circle is the distance from the center to any point on the circle. The first circle in the Explore! has a radius of 3 inches. The second circle has a radius of 2 centimeters. center radius A chord is a line segment having both endpoints on the circle. Chords not Chords 114 Lesson 20 ~ Parts Of Circles

4 A diameter is a special chord. It has endpoints on the circle, but also goes through the center of the circle. The diameter is the distance across a circle through its center. It is twice as long as a radius. diameter example 1 solutions a. The diameter of a circle is 10 meters. Find its radius. b. The radius of a circle is 3 inches. Find its diameter. a. A radius is half the length of a diameter. Find the radius by dividing the diameter by = 5 The radius is 5 meters. b. A diameter is twice as long as a radius. Find the diameter by multiplying the radius by = 6 The diameter is 6 inches. 10 m 3 in A central angle in a circle is an angle that has its vertex at the center of the circle. Angles are measured using degrees. Each circle has 360 around its center point. There are two central angles shown at the left. example 2 Find the measure of angle solution Add the two given angles = 240 There are 360 in a circle. To find the measure of angle 1, calculate the difference between 240 and = 120 The measure of angle 1 is 120. This can also be written m 1 = 120. example 3 Find the probability that a dart landing on the circle will land in the shaded part. solution P(lands in the shaded part) = degrees shaded total degrees = = _ The probability that a dart lands in the shaded part is 1_ 3 or 33 1_ 3 % or 0. _ 3. Lesson 20 ~ Parts Of Circles 115

5 exercises Identify each part of the circle drawn in red as a radius, chord, diameter, central angle or none of these Sketch a circle with a radius of 1 inch. Draw a radius on your sketch. 11. Sketch a circle with a diameter of 4 centimeters. Draw a diameter on your sketch. 12. Sketch a circle. Draw a chord that is not a diameter. 13. Sketch a circle. Draw a central angle that is Sketch a circle. Draw a central angle that is 180⁰. What else could this angle be called in the circle? given the radius, find the diameter of each circle ft mm cm _ 4 m yd in given the diameter, find the radius of each circle cm yd mm _ 4 in ft _ 3 yd Find the degree measurement of angle 1 in each circle Lesson 20 ~ Parts Of Circles

6 Find the probability that a dart that randomly lands on the circle will land in the shaded area ⁰ ⁰ 35. Draw and shade a central angle of a circle so a dart landing on the circle will have a 1_ 4 chance of landing in the shaded area. What is the measure of the shaded central angle? 36. Draw and shade a central angle of a circle so a dart landing on the circle will have a 1_ 3 chance of landing in the shaded area. What is the measure of the shaded central angle? review Convert each fraction to a decimal _ _ 5 _ _ round each number to the nearest hundredth _ _ 6 Lesson 20 ~ Parts Of Circles 117

7 tic-tac-toe ~ missing Angles There are 360 around the center point of a circle. This means the central angles inside a circle must all sum to 360. Example: Two missing central angles are shown in red. Find m 1. Notice that 1 and the 90 angle make half the circle. 1_ 360 = m 1 = = 90 m 1 = 90 Find m 2. Notice that 2 and the 60 angle make half the circle which is 180. m 2 = = 120 You can find m 2 if you know m 1 because all four angles sum to 360. m 2 = = 120 m 2 = 120 Find the missing central angles in each circle below. The missing central angles are shown in red. Show all work. A percent or fraction outside the circle shows the portion of the circle represented by the central angle % _ 1_ 3 40% _ Lesson 20 ~ Parts Of Circles

8 circumference OF a circle Lesson 21 To find the perimeter of a shape, you find the sum of the lengths of its sides. Circles do not have sides because they are not made up of line segments. The distance around a circle is called its circumference. You will look for the relationship between the circumference of a circle and its diameter in this Explore! explore! round and round step 1: Wrap a piece of string around a circular object one time. Measure the length of string that fits around the object using centimeters. This is the circumference of the circular object. step 2: Measure the diameter of the circular object using centimeters. step 3: Copy the table below. Record your measurements from steps 1 and 2. Find the ratio of the circumference of the circle to its diameter. Use a calculator to write this ratio as a decimal to the nearest hundredth. Circular object Circumference diameter Circumference diameter step 4: Repeat steps 1 and 2 for four more objects. Use a calculator each time to find the ratio of the circumference of the circle to its diameter to the nearest hundredth. Record your results in the chart. step 5: What do you notice about each ratio of circumference to diameter? step 6: The circumference of a circle is about times as long as its diameter. step 7: Estimate the circumference of a circle with a diameter of 5 inches. The circumference of a circle is a little bit more than three times the length of the diameter. You can wrap 3 diameters along the edge of a circle. There will still be a little bit of the circle not covered because the circumference is larger. 1 diameter 3 diameters 2 diameters Lesson 21 ~ Circumference Of A Circle 119

9 The exact number of times the diameter can be wrapped around the circle is represented by the Greek letter π (pi). Pi is the ratio of the circumference of a circle to its diameter. The exact value of π cannot be written as a decimal because it never terminates and never repeats. Most people round π to the nearest hundredth and use the number 3.14 or the fraction 22 to estimate π. 7 example 1 Find the circumference of each circle. use 3.14 for π. a. diameter = 8 ft b. radius = 6.5 m 8 ft 6.5 m solutions a. Write the formula. Circumference = π diameter Use 3.14 for π and 8 for the diameter. Circumference Multiply = The circumference is about ft. b. The diameter is twice the radius = 13 m Use 3.14 for π and 13 for the diameter. Circumference Multiply = The circumference is about m. example 2 solution trina made a garden decoration by shaping a piece of metal into a circle. she wanted a circular ornament with a diameter of 6 inches. About how long was the original piece of metal in order to make the outside of the ornament? The length of the metal is the circumference of the circle. Write the formula. Circumference = π diameter Use 3.14 for π and 6 for the diameter. Circumference Multiply = The original piece of metal for the garden decoration was about inches. 120 Lesson 21 ~ Circumference Of A Circle

10 exercises Find the circumference of each circle. round each answer to the nearest tenth. use 3.14 for π cm 4 cm 50 ft in 1 1_ 2 m 200 mi 7. diameter = 2 in 8. diameter = 4.2 mm 9. radius = 2.5 ft 10. Copy the table below. a. Find the circumference of a circle with a diameter of 5 ft using the three different values for π listed below. Round each answer to the nearest ten thousandth (4 numbers after the decimal point). Use the π key on your calculator for the last circumference. Circumference = π diameter Circumference 3.14 Diameter Circumference 22 Diameter 7 Circumference = π diameter b. Which circumference is the smallest? Which circumference is the largest? c. Does the value of π used really affect your answer? Explain. 11. Pat and Donna found the circumference of a circle with a radius of 5 in. Their work is shown below. Who was correct? Explain. Pat Circumference The circumference is about 15.7 in. donna Circumference The circumference is about 31.4 in. 12. Draw and label the diameter of a circle whose circumference is more than 4 feet and less than 8 feet. 13. The diameter of a circle is 6 ft. Would its circumference be greater than, less than or equal to 18 ft? Explain how to determine this answer without using a calculator. 14. The radius of a circle is 5 cm. Would its circumference be greater than, less than or equal to 30 cm? Explain how to determine this answer without using a calculator. Lesson 21 ~ Circumference Of A Circle 121

11 15. The Ferris wheel at the waterfront during the Rose Festival has a diameter of 50 ft. Find the distance a person travels in one revolution. Round to the nearest foot. 16. A circular track around Jan s school has a radius of 200 ft. Find the distance she jogs in one lap. 17. A pivoting sprinkler line at the Collins farm in Enterprise, Oregon is 1_ 4 of a mile long. Find the length of the outer edge of the field watered in one rotation. 1_ 4 mi 18. The diameter of a circle can be found if its circumference is known. To find the circumference of a circle, multiply the diameter by π. To find the diameter of a circle, divide the circumference by π. diameter = circumference π a. Find the diameter of a circle with a circumference of 314 m. Use 3.14 for π. b. Find the diameter of a circle with a circumference of 47.1 in. Use 3.14 for π. 19. Julie found the circumference of a circle with a diameter of 10 ft. She used 3.14 for π. She concluded that the circumference is exactly 31.4 ft. Is she correct? Explain why or why not. review 20. Draw a circle with a radius of 2 in. 21. Draw a circle with a diameter of 6 cm. 22. Sketch a circle. Draw a chord that is not the diameter of the circle. Find the probability that a dart that randomly lands on the circle will land in the shaded area Lesson 21 ~ Circumference Of A Circle

12 tic-tac-toe ~ Arc length Paulson repaired a section of fencing around his circular patio. He measured the central angle enclosing the arc. He also found the radius of his patio. His measurements are shown to the right. The red length on the circle is the section he repaired. How much fencing did Paulson buy? An arc on a circle is a piece of the outer edge of the circle. It is a piece of the circumference. You can find an arc s length if you know the diameter of the circle and the central angle that encloses the arc m Example: Find the length of the red arc by using the steps below. step 1: Find the circumference of the circle. step 2: Find the ratio of the central angle degree to the total degrees (360 ). step 3: The arc is 1_ 4 of the circumference. Find the arc length = m = 1_ 4 1_ = = m 4 Paulson bought about m of fencing. Find the arc length for each circle below in 50 m ft yd cm mm Lesson 21 ~ Circumference Of A Circle 123

13 tic-tac-toe ~ π π There are two common approximations for π. One is The other is Investigate π and its approximations using a spreadsheet. Note: The * symbol represents multiplication in spreadsheets. PI( ) represents π. hint: More decimal places can be displayed in cells. Highlight the desired cells, go to Format and select cells. Select numbers and change the number of decimal places on the right. step 1: Open a spreadsheet on a computer. Create column headings for your spreadsheet. In cell B1 type: Circumference Using 3.14 In cell C1 type: Circumference Using 22 7 In cell D1 type: Circumference Using Pi step 2: Place the lengths of the diameters in column A. The lengths should start with 1 unit. Multiply by 10 for each following diameter. Continue listing diameters that are multiples of 10 in column A until you reach 1,000,000 units. step 3: Type =3.14*A2 in cell B2. Remember: circumference = π diameter. This calculates the circumference of the circle using 3.14 times the diameter in cell A2. Place the cursor in the lower right hand corner of cell B2. A small black box appears. Click, hold the mouse and drag the small black box down to cell B8. The spreadsheet uses the diameter in each A cell to calculate the circumference using step 4: Type =(22/7)*A2 in cell C2. This calculates the circumference of the circle using 22 7 times the diameter in A2. Place the cursor in the lower right hand corner of the C2 cell as you did in step 3. Click, hold the mouse and drag the small black box down to cell C8. The spreadsheet uses the diameter in each A cell to calculate the circumference using step 5: Type =PI( )*A2 in cell D2. This calculates the circumference of the circle using π times the diameter in A2. Click on the D2 cell. As in step 3, click and drag the box down to the D8 cell. step 6: Print your spreadsheet. Answer the following questions on a separate sheet of paper. Attach it to your spreadsheet. 1. Order the numbers π, 3.14 and 22 7 from least to greatest. 2. What is the difference between the circumference of a circle with a diameter of 1,000,000 units using π versus using 3.14? 3. What is the difference between the circumference of a circle with a diameter of 1,000,000 units using π versus using 22 7? 4. Scientists and mathematicians prefer to use π instead of the common approximations. Why do you think this is? Describe at least one situation where using an approximation could cause problems in a real-world situation. 124 Lesson 21 ~ Circumference Of A Circle

14 Perimeter and circumference Lesson 22 It is possible to find the perimeter of a shape made of a combination of rectangles, triangles and circles by using perimeter and circumference. Look at the figure to see the combination of shapes that make up the whole figure. explore! FlOwer garden Darcy put a fence around her flower garden. Her garden is the shape of a rectangle and a half circle put together as shown. She put the fence along the outer edge (black line) but not along the red line. Find the length of fencing she needed. step 1: Find the length of fence she needed around the 3 sides of the rectangle. step 2: Find the diameter of the half circle. 8 ft step 3: Imagine the half circle was a full circle. Find the circumference of the full circle. step 4: How much fencing was needed for half of the circle? 12 ft step 5: How much total fencing did Darcy need for the fence around the entire garden? example 1 Find the perimeter of each shape. a. b. 6 in 4 m 4 in 3 in 5 in solutions The perimeter equals the total length of all outside edges of the shape. a. This is a half circle. Find + 4 m Find the circumference of the whole circle = Find half the circumference for the half-circle = 6.28 The perimeter is the half-circle circumference plus the length of the diameter = The perimeter is about meters. 6 in 4 in 5 in + 3 in b. This is a rectangle and a triangle. Find 6 in Find the perimeter around the rectangle = 16 in Find the perimeter around the triangle = 8 in Add the two perimeters together = 24 in The perimeter is 24 inches. Lesson 22 ~ Perimeter And Circumference 125

15 You can find the length of a missing side when you know the perimeter of the shape. example 2 Find the height of the rectangle. The perimeter of the shape is 34 feet. 5 ft 10 ft 5 ft solution Find the perimeter you know. The bottom of the rectangle is 10 ft so the top is also 10 ft. Add all sides to find the known part of the perimeter. The total perimeter is 34 ft. Find the difference between 34 and 30. The height of the rectangle is 4 ft = 30 ft = 4 ft exercises Find each stated missing length. 1. diameter of half circle 2. base of rectangle 3. base of square 6 mm 10 ft 5 in 12 in 13 in 6 mm Find the perimeter of each shape. 3 m in 10 in 3.5 cm 3.5 cm 4 cm 4 cm 2.5 m 5 m 2.5 m 7. 2 in ft 6 ft 6 mm 15 mm 7 in 7 in 6 ft 126 Lesson 22 ~ Perimeter And Circumference

16 yd 5 km 5.3 yd 14.5 km 1 m 2 m 2.5 m 2.5 m 13. Find the height of the rectangle. The perimeter of the shape is 24 ft. 6 ft 5 ft 4 ft 14. Find the base of the rectangle. The perimeter is 20 m. 2.5 m 2.5 m 2 m 15. Kim and Maria found the perimeter of the shape at right. Who found the correct perimeter? Explain your answer. Kim = = The perimeter is about in. Maria = = The perimeter is about in. 9 inches review 16. Shaquina bought a sweater. The original price was $40. It was on sale for 40% off. Find the discounted cost of the sweater. 17. Mario took his parents out to lunch. The meal cost $22. Mario left a 15% tip on the table. How much was the tip? 18. Terry is 20% taller than his brother, Nathan. Nathan is 50 inches tall. How tall is Terry? Find the probability that a dart randomly hitting the circle will land in the shaded area Lesson 22 ~ Perimeter And Circumference 127

17 tic-tac-toe ~ circumference Measure the diameter of a car tire and a bike tire to the nearest tenth of a centimeter. Find the circumference of each one. Diameter Circumference Car Tire Bike Tire Wrap a piece of string around a basketball one time. Place the string along a ruler and record the length to the nearest tenth of a centimeter. This is the circumference of the basketball. Find the diameter of the basketball from its circumference. Repeat this process to find the diameter of two different balls. Identify the ball circumference and diameter in the chart. Circumference Diameter Basketball?? Find one other circular object at home. Record the object, its circumference and its diameter. tic-tac-toe ~ mural Your art teacher wants to create a mural on the hallway wall outside the library. She has a 10 ft by 12 ftrectangular space on which to work. She wants four composite painted figures in the mural. She will also outline the composite figure with glitter rope. Your class will create the mural with four composite figures and find the length of glitter rope needed. The composite shapes must include: two rectangles. a rectangle and a half-circle. 12 ft a rectangle and a triangle. one of your choice (there are examples in Lesson 22). 10 ft Turn in: a sketch of your mural with the actual dimensions labeled. organized work for each perimeter below your sketch or on a separate sheet of paper. the total length of glitter rope needed to outline all shapes. 128 Lesson 22 ~ Perimeter And Circumference

18 similar and congruent Figures Lesson 23 explore! match the shapes step 1: Match the pairs of figures below that have the exact same shape. They may or may not be the same size. A B n g J C M e L d F h I K step 2: Which pair(s) of shapes are also the same size? step 3: Sketch another figure that is the same shape of each matched pair but a different size. Figures with the exact same shape are called similar figures. Similar figures do not necessarily need to be the same size. similar not similar Figures with the same shape and the same size are congruent figures. Congruent figures are special similar figures. Examples of congruent figures. Lesson 23 ~ Similar And Congruent Figures 129

19 example 1 determine whether each pair of figures is similar, congruent or neither. a. b. c. solutions a. These circles are similar. They are the exact same shape, but not the same size. b. These figures are neither. They are not the same size nor the same shape. c. These figures are congruent. They are the same size and shape. The parts of similar figures that match are corresponding parts. Look at ABCD and EFGH below. A B e F d AB corresponds to EF BC corresponds to FG CD corresponds to GH DA corresponds to HE C h g A corresponds to E B corresponds to F C corresponds to G D corresponds to H example 2 solutions dog and CAt are similar triangles. a. List the pairs of corresponding sides. b. List the pairs of corresponding angles. c. Find the ratio of each side in dog to the corresponding side in CAt. a. DO corresponds to CA OG corresponds to AT GD corresponds to TC t 8 m A d b. D corresponds to C 5 m O corresponds to A 3 m 10 m 6 m G corresponds to T o 4 m g C c. Ratio of DO to AC is 3_ 6 = 1_ 2 or 1 : 2. Ratio of OG to AT is 4_ 8 = 1_ 2 or 1 : 2. Ratio of GD to TC is 5 10 = 1_ 2 or 1 : 2. When two figures are similar, the ratios of their corresponding sides are equal. The ratio 1 : 2 means CAT is twice as large as DOG in example 2. This also means DOG is half the size of CAT. 130 Lesson 23 ~ Similar And Congruent Figures

20 exercises determine whether each pair of figures is similar, congruent or neither Are all circles similar? Explain using words and/or pictures. 8. Are all squares similar? Explain using words and/or pictures. 9. Are all rectangles similar? Explain using words and/or pictures. 10. Are all triangles similar? Explain using words and/or pictures. sketch a figure similar, but not congruent, to each shape sketch a figure congruent to each shape s 15. CAR is similar to SUV. a. List the corresponding sides. b. List the corresponding angles. c. Find the ratio of each side in CAR to the corresponding side in SUV. C 5 in A 8 in 6 in 10 in u r 16 in 12 in V 16. ROSE is similar to PINK. a. List the corresponding sides. b. List the corresponding angles. c. Find the ratio of each side in ROSE to the corresponding side in PINK. r e 15 cm o 9 cm s P K 5 cm I 3 cm n Lesson 23 ~ Similar And Congruent Figures 131

21 17. ORE is congruent to LCK. Find the ratio of each side in ORE to the corresponding side in LCK. O 20 ft 29 ft L 20 ft 29 ft R 21 ft E C 21 ft K 18. READ is congruent to MATH. a. Find the perimeter of READ. b. Find the perimeter of MATH. c. When two figures are congruent, their perimeters are. R 2 in 2 in A E 4.5 in 4.5 in M 2 in 2 in A 4.5 in 4.5 in T I 2 cm 3 cm D 19. Sketch YOU so it is similar to IME shown at left. The ratio of corresponding sides of YOU to IME is 1 : 2. H E 4 cm M 20. Sketch two rectangles that are similar and have a ratio of corresponding sides of 1 : 3. Label the lengths of the sides. review Find the perimeter of each figure in 5.7 in 10 in 10 in 4 in 24. Sketch a 120 central angle in a circle. 25. The diameter of a circle is 15 m. Find the radius. 9 m 20 m tic-tac-toe ~ Pictures Find three pairs of congruent figures and three pairs of similar figures that are not congruent. The figures must be polygons. You can take photos or cut pictures out of magazines. Once you have your pictures, place them on a poster. Label the corners (vertices) of the figures with capital letters. List all pairs of corresponding sides. List all pairs of corresponding angles. Record these on the poster next to the appropriate pictures. Measure corresponding sides. Write the ratio of corresponding sides between the figures. Record these on the poster next to the appropriate pictures. 132 Lesson 23 ~ Similar And Congruent Figures

22 ratios and similar Figures Lesson 24 explore! ratio OF lengths and Perimeters step 1: Use the similar squares to the right. a. Find the ratio of each pair of corresponding sides from square A A to square B. B 2 in b. Find the perimeter of each square. c. Find the ratio of the perimeters of square A to square B. 5 in d. What do you notice about the ratio of the corresponding sides and the ratio of the perimeters? step 2: Use the similar circles to the right. a. Find the ratio of the diameter in circle C to the diameter in circle D. b. Find the circumference of each circle. Use 3.14 for π. c. Find the ratio of the circumference of circle C to the circumference of circle D. Write the ratio without decimals. d. What do you notice about the ratio of the corresponding diameters and the ratio of the circumferences? 1 m C 3 m D step 3: Two rectangles are similar. The ratio of their corresponding sides is 1 : 5. The perimeter of the smaller rectangle is 20 inches. Predict the perimeter of the larger rectangle. Explain how you came up with your answer. When two figures are similar, the ratio of each pair of corresponding lengths is equal. No matter which two corresponding sides, diameters, radii, or even perimeters you find the ratio of, each corresponding pair in a shape will have the same ratio. example 1 two similar figures are shown below. Find the ratio of the smaller figure s perimeter to the larger figure s perimeter. 4 ft 8 ft solution The ratio of the perimeters is the same as the ratio of the sides since the shapes are similar. Find the ratio of the sides. The ratio of the perimeters is 1_ 2 or 1 : 2. 4_ 8 = 1_ 2 Lesson 24 ~ Ratios And Similar Figures 133

23 The equal ratios for lengths in similar figures can be used to find missing sides and missing perimeters. example 2 given the similar figures, use ratios to find the missing perimeter. 3 in 9 in solution Perimeter = 12 in Perimeter =? in The ratio of the perimeters equals the ratio of any pair of corresponding sides. Find the ratio of the corresponding sides. 3_ 9 = 1_ 3 Find the ratio of the perimeters. 12? 12 Set these ratios equal and use equivalent fractions to find the larger perimeter. The perimeter of the larger triangle is 36 in. 1_ 3 = 12? 1_ 3 = example 3 given the similar figures and perimeters, use ratios to find the length of the missing side.? ft 4 ft Perimeter = 10 ft Perimeter = 25 ft solution The ratio of the perimeters equals the ratio of any pair of corresponding sides. Find the ratio of the perimeters. Find the ratio of the corresponding sides. Set these ratios equal and use equivalent fractions to find the larger perimeter. The corresponding side of the larger triangle is 10 ft = 2_ 5 4_? 2_ 5 = 4_? 2_ 5 2 = Lesson 24 ~ Ratios And Similar Figures

24 exercises Find the ratio of the perimeters or circumferences in each pair of similar figures below. Compare the left shape to the right shape in 1 ft 5 in 4 ft 1 m 2 m cm 8 yd 9 ft 6 cm 32 yd 21 ft The perimeters or circumferences for each pair of similar figures are shown. Find the ratio of the corresponding sides or diameters Perimeter = 20 ft Perimeter = 12 in Perimeter = 60 ft Circumference 9 m Circumference 15 m Perimeter = 24 in use ratios to find the missing perimeter or circumference for each pair of similar figures in Perimeter = 12 in 6 ft 3 m 4 in Perimeter =? in Perimeter = 16 ft 15 ft Perimeter =? ft Perimeter = 15 m 2 m Perimeter =? m Lesson 24 ~ Ratios And Similar Figures 135

25 use ratios to find the missing corresponding side length or diameter for each pair of similar figures. 3 cm ? m 15. Perimeter = 8 cm Perimeter = 13 m? cm 15 m Perimeter = 16 cm 1 in C 3.14 in? in C in 16. Explain what a ratio of 1 : 4 means if it is the ratio of corresponding sides of two similar rectangles. 17. Explain what a ratio of 1_ 3 means if it is the ratio of corresponding perimeters of two similar triangles. 18. Draw a square with side lengths of 2 cm using a ruler. Draw a square with sides twice as long as the sides of the first square. a. What are the side lengths of the second square? b. What is the ratio of the perimeters of the first square to the second square? 19. Two similar rectangles have a ratio of corresponding sides of 7 : 8. Find the ratio of their perimeters. 20. The ratio of perimeters of two similar triangles is 2 : 5. The smaller triangle has a base of 6 in. Find the base of the larger triangle. 21. Sketch two similar rectangles that have perimeters with a ratio of 2 : 3. Label the measurements of the base and height of each rectangle. 22. Two figures are congruent. The ratio of their perimeters is :. Explain your answer. review Perimeter = 26 m 23. LAK is similar to TRE. a. List the corresponding sides. b. List the corresponding angles. c. Find the ratio of each pair of corresponding sides. L 3 cm A 6 cm 4 cm T K 9 cm 18 cm Find the perimeter of each figure. R 12 cm E cm 8.9 cm 6 in 4 cm 4 cm 6 cm 4 m 10 m 136 Lesson 24 ~ Ratios And Similar Figures

26 review BLoCK 4 vocabulary center circumference diameter central angle congruent figures π (pi) chord corresponding parts radius circle similar figures Lesson 20 ~ Parts of Circles Identify each part of the circle drawn in red as a radius, chord, diameter, central angle or none of these Find the diameter of a circle with each given radius. 6. Find the radius of a circle with each given diameter. a. 12 cm a. 8 ft b. 4.5 in b. 5 mm c. 2 1_ 4 m c. 6 1_ 2 km Find the degree measurement of angle 1 in each circle Find the probability that a dart randomly landing on each circle will land in the shaded area ⁰ ⁰ Block 4 ~ Review 1 37

27 Lesson 21 ~ Circumference of a Circle Find the circumference of each circle. use 3.14 for π cm 2 ft 1.5 in 16. diameter = 5 cm 17. diameter = 2.4 mm 18. radius = 6.5 ft 19. A circular track around Tran s school has a radius of 300 ft. Find the distance he walks during each lap around the track. Lesson 22 ~ Perimeter and Circumference Find the perimeter of each shape in 8 in 8 in 30 in 1.25 m 1.2 m 2 in 2.5 m 2.5 m 3 m 5 m cm cm 15 cm 15 cm 6 mm 8 mm 25. Find the height of the rectangle. The perimeter of the shape is 48 ft. 12 ft 10 ft 6 ft 138 Block 4 ~ Review

28 Lesson 23 ~ Similar and Congruent Figures determine whether each pair of figures is similar, congruent or neither Sketch a figure similar to the one shown. 30. Answer always, sometimes or never. a. Circles are similar. b. Squares are congruent. c. Rectangles are similar. 31. ROAM is similar to WEST. a. List the corresponding sides. b. List the corresponding angles. c. Find the ratio of each side in ROAM to the corresponding side in WEST. R O W 4 cm M 8 cm A T 2 cm E 1 cm S 32. TIM is congruent to JON. a. List the corresponding sides. b. List the corresponding angles. c. Find the ratio of each side in TIM to the corresponding side in JON. T 11 ft M 21 ft 12 ft I J 11 ft N 21 ft 12 ft O Lesson 24 ~ Ratios and Similar Figures Find the ratio of the perimeters or circumferences in each pair of similar figures below. Compare the left shape to the right shape in 1 m 6 in 5 cm 3 m 10 cm The perimeters or circumferences for each pair of similar figures are shown. Find the ratio of the corresponding sides or diameters Perimeter = 27 ft Perimeter = 45 ft Circumference 15 m Circumference 6 m Block 4 ~ Review 139

29 use ratios to find the missing perimeter or circumference for each pair of similar figures ft 6 ft Perimeter = 10 ft Perimeter =? ft 3 m Circumference =? m 1 m Circumference = 3.14 m use ratios to find the missing corresponding side length for each pair of similar figures cm? cm 41. Perimeter = 8 cm Perimeter = 16 cm 6 m Perimeter = 12 m? m Perimeter = 48 m 42. The ratio of perimeters of two similar rectangles is 3 : 5. The smaller rectangle has a base of 9 in. Find the base of the larger rectangle. tic-tac-toe ~ GaMe Board step 1: Create a circular game board with a 1_ 4 chance that a randomly thrown dart will land in the shaded part of the circle. What is the degree measure of the central angle that is shaded? step 2: Create another circular game board that still has a 1_ 4 chance of landing in the shaded part. It should have two shaded central angles which are not touching each other (except at the center of the circle). The shaded portions do not have to be the same size. What are the degree measures of the two central angles you shaded? step 3: Create another circular game board that still gives a 1_ 4 chance of landing in the shaded part. It should have three central angles shaded which are not touching each other (except at the center of the circle). The shaded portions do not have to be the same size. What are the degree measures of the three central angles you shaded? step 4: Repeat #1 3, but this time a dart needs to have a 2_ 3 chance of landing in the shaded area. 140 Block 4 ~ Review

30 tic-tac-toe ~ haiku A haiku poem is a three line poem. The first line is five syllables. The second line is seven syllables. The third line is five syllables. Example: A haiku poem about similar figures. Similar Figures Exact same shape, change in size Sometimes equal size Write a haiku poem for each of the following parts of a circle. radius diameter chord central angle circumference When you are finished, you will have five haiku poems. Put the five poems in a booklet. Add illustrations if desired. tic-tac-toe ~ VocABulAry QuiZ Create a vocabulary study guide for students with the vocabulary words from this textbook. The study guide should include each word s definition and a picture or example illustrating the word. The study guide may be typed or hand-written. Create a quiz, on a separate sheet of paper, using at least 10 of the vocabulary words from your study guide. The quiz should include a matching section and fill-in-the-blank section. Turn in the vocabulary list, the quiz and the answers to the quiz. Block 4 ~ Review 141

31 PAtti BAnk ceo Bend, oregon CAreer FoCus I am the CEO of a bank. I help oversee our company and part of my job is working with our board of directors to determine what will be best for our bank. I also work with investors who decide whether or not they should make investments in our stock. Overall, I manage all employees and each department in our company. I use math in almost everything I do. About half of my work is done with numbers. I review all bank financial statements to make sure we are making money. I also make sure our investments are safe and will continue to make money in the long term. Much of my job requires predicting what will happen in the future. There are many formulas and mathematical concepts that help me make those predictions. It would not be possible to do my job without a good understanding of math. To get a job as a CEO usually requires having a college degree. My job requires many different skills. Studying business or accounting is most helpful. Communication is also an important part of what I do. Studying how to communicate effectively is beneficial. Bank employees make a wide range of salaries. Clerks typically earn the least, while a CEO will typically earn the most. There is a broad range of jobs between teller and CEO. The salaries differ for each position. The best part of my job is figuring out how to best run a business. An understanding of numbers helps me make important decisions. I like the challenge of putting all of the pieces together. My profession gives me the opportunity to work with others on a daily basis, which is something I enjoy. 142 Block 4 ~ Review