9A Ready To Go On? Skills Intervention 9-1 Developing Formulas for Triangles and Quadrilaterals Finding Measurements of Parallelograms Find each measurement. A. the area of the parallelogram A b Use the formula for area of a parallelogram. A ( ) Substitute for b and A in. 2 Simplify. for h. The area of the parallelogram is in. 2 B. the height of the rectangle in which A (6 x 2 2x) f t 2. 7 cm A b Use the formula for area of a rectangle. h Substitute 6 x 2 2x for A and for b. 2x( ) 2x(h) Factor 2x out of the expression for A. 2x(3x 1) 2x(h) Divide both sides by. 15 cm 12 cm 2x ft h The height of the rectangle is Simplify. ft. Finding Measurements of Rhombuses and Kites Find d 1 of a kite in which d 2 6 yd and A 42 y d 2. A 1 2 ( ) Use the formula for area of a kite. 1 2 ( d 1 )( ) Substitute for A and d 2 = 6 yd d 1 42 d 1 Simplify. 42 3 d 1 for d 2. Divide both sides of the equation by. d 1 What is d 1? 125 Holt Geometry
9A Ready To Go On? Problem Solving Intervention 9-1 Developing Formulas for Triangles and Quadrilaterals You can use the Pythagorean Theorem to find the height of a parallelogram. The decorative tile shown is made up of 1 in. squares. Use the grid to find the perimeter and area of the decorative tile. Understand the Problem 1. What shape is the decorative tile? 2. What are you being asked to do? 3. What are the dimensions of the grid? Make a Plan 4. State the Pythagorean Theorem. 5. State the formula for the area of a triangle. 6. Find the area of each triangle outside of the tile. A 1 A 2 A 3 A 4 7. What is the area of the rectangular grid? Solve 8. Use the Pythagorean Theorem to find the side lengths of the parallelogram. 2 2 4 2 and 2 2 2 2 C 1 2 5 in. C 2 2 2 in. 9. What is the perimeter of the parallelogram? 10. What is the area of the tile? Look Back 11. How is the area of the tile related to the area of the grid? 12. Does the tile look like it covers about half of the grid? 13. Do your answers to Exercises 9 and 10 seem reasonable? 126 Holt Geometry
9A Ready To Go On? Skills Intervention 9-2 Developing Formulas for Circles and Regular Polygons Find these vocabulary words in Lesson 9-2 and the Multilingual Glossary. Vocabulary circle center of a circle center of a regular polygon apothem central angle of a regular polygon Finding Measurements of Circles Find each measurement. A. the area of A in terms of. What is the radius of A? A Use the formula for area of a circle. A ( ) 2 Substitute for r. A f t 2 Simplify. B. the circumference of A in terms of. C Use the circumference of a circle. C 2 ( ) Substitute for r. C ft Simplify. A 28 ft Finding the Area of a Regular Polygon Find the area of a regular hexagon with side length 18 in. Round to the nearest tenth. Find the perimeter of the hexagon. P 6( ) in. The hexagon can be divided into with side length 18 in. By the 30-60 -90 Triangle Theorem, the apothem is in. Write the formula for area of a regular polygon. In the area formula, a represents the and P represents the. equilateral triangles of the polygon, A 1 2 Substitute for a and for P. A 3 Simplify. A in. 2 9 in. 18 in. 127 Holt Geometry
9A Ready To Go On? Skills Intervention 9-3 Composite Figures Find this vocabulary word in Lesson 9-3 and the Multilingual Glossary. Vocabulary composite figure Finding the Area of Composite Figures by Adding Find the shaded area. Round to the nearest tenth, if necessary. 2 m Divide the figure into two rectangles and a trapezoid. Fill in the missing side lengths on the figure. 3 m 6 m Find the area of the rectangles, using A bh. A ( ) ( ) m 2. A ( ) ( ) m 2. 8 m 2 m Find the area of the trapezoid, using A 1 2 (b 1 b 2 )h. A 1 2 ( )( ) m 2. Add the areas of the simple shapes to find the area of the composite figure. m 3 m m 6 A m 2. m m m The area of the figure is m 2. Estimating Areas of Irregular Shapes Use a composite figure to estimate the shaded area. The grid has squares with side lengths of 1 cm. Draw a composite figure that approximates the irregular shape. Find the area of each part of the composite figure. area of triangle I: A 1 2 bh 1 2 c m 2. area of trapezoid II: A 1 2 ( b 1 b 2 )h 1 2 c m 2. area of triangle III: A 1 2 bh 1 2 c m 2. Add the areas of the simple shapes to estimate the area of the irregular shape. A. The shaded area is about c m 2. II III I 128 Holt Geometry
9A Ready To Go On? Problem Solving Intervention 9-3 Composite Figures To find the area of a composite figure, find the sum of all of the simple shapes that make up the composite figure. Howard is planting sod on an irregularly shaped garden as shown. The grid has squares with side lengths of 1 m. Estimate the area of the garden. Given that sod cost $5.99 per square meter, find the cost of the sod. Understand the Problem 1. What two problems must you solve? 2. Why can you not find the exact area? Make a Plan 3. What is the purpose of drawing a composite figure that separates the shape into three regions? a b 4. After finding the area of the composite figure, how will you estimate the cost of the sod? c Solve 5. Find the area of triangle a. 6. Find the area of trapezoid b. 7. Find the area of triangle c. 8. Find the area of the composite figure. 9. The area of the garden is approximately. 10. Calculate the approximate cost of the sod. Look Back 11. Estimate the area of the irregular shape by subtracting the regions outside the composite figure from the area of the grid, 36 m 2. 12. Does your answer in Exercise 11 match your answer in Exercise 9? 129 Holt Geometry
9A Ready To Go On? Quiz 9-1 Developing Formulas for Triangles and Quadrilaterals Find each measurement. 1. the area of the parallelogram 2. the base of the rectangle, in which A (14 x 2 35x) f t 2 6 cm 9 cm 7x ft 4 cm 3. d 1 of the kite, in which A 96 in. 2 4. the area of the rhombus d 2 = 16 in. d 1 24 m 13 m 5. Part of a stained glass window is shown at the right. Each square in the grid is 1 in. long. Find the perimeter and area of the dark gray parallelogram. 9-2 Developing Formulas for Circles and Regular Polygons Find each measurement. Round to the nearest tenth. 6. the circumference of C in terms of 7. the area of K in terms of C 8 yd K 8x m 130 Holt Geometry
9A Ready To Go On? Quiz continued Find the area of each regular polygon. Round to the nearest tenth. 8. a regular hexagon with side length 8 cm 9. a regular pentagon with apothem 10 m 9-3 Composite Figures Find the shaded area. Round to the nearest tenth, if necessary. 10. 11. 16 ft 4 cm 5 ft 12. Joel is fertilizing an irregularly shaped garden, as shown. The grid has squares with side lengths of 1 m. Estimate the area of the garden. Given that fertilizer cost $2.99 per square meter, find the cost of the fertilizer. 131 Holt Geometry
9A Ready To Go On? Enrichment Developing Geometric Formulas 1. The circumference of a circle is equal to the area of the circle. Find the radius. Match each figure to its area. Areas have been rounded to the nearest tenth. 2. 84.8 cm 2 A 18 cm 29 3. 90 c m 2 B 11 cm 4. 92.7 c m 2 C 3 cm 5. 95 c m 2 D 8 cm 12 cm 75 132 Holt Geometry
9B Ready To Go On? Skills Intervention 9-4 Perimeter and Area in the Coordinate Plane Finding Perimeter and Area in the Coordinate Plane Draw and classify the polygon with vertices P ( 5, 5), Q (1, 1), R ( 1, 2), and S( 7, 2). Find the perimeter and area of the polygon. Step 1 Draw the polygon. Step 2 PQRS appears to be a rectangle. To verify this, use slopes to show that the sides are perpendicular. slope of _ PQ : slope of _ QR : 1 ( 5) 2 1 3 2 8 6 4 6 4 2 2 2 2 y x slope of _ RS : 2 ( 1) 6 slope of _ 5 SP : ( 7) The consecutive sides are perpendicular so PQRS is a. Step 3 Let _ RS be the base of the rectangle and _ RQ be the height. Use the Distance Formula to find each side length. Write the Distance Formula: b RS 7 2 2 2 2 2 2 h RQ 1 2 1 2 2 2 The perimeter is P 2b 2h 2 2 units. The area is A bh ( )( ) 2( ) 26 unit s 2. 133 Holt Geometry
9B Ready To Go On? Skills Intervention 9-5 Effects of Changing Dimensions Proportionally Effects of Changing Dimension Proportionally Describe the effect of each change on the perimeter and area of the given figure. A. The base and height of the rectangle are both multiplied by 4. Original dimensions of the figure: P 2b 2h 2( ) 2( ) m A bh ( )( ) m 2 When dimensions are multiplied by 4: P 2b 2h 2( ) 2( ) m A bh ( )( ) m 2 2 m 8 m 5 m 20 m The perimeter is multiplied by. The area is multiplied by 2 or. B. The radius of P is multiplied by 1 2. Original dimensions: Dimensions multiplied by 1 2 : C 2 r 2 ( ) in. C 2 r 2 ( ) in. A r 2 ( ) 2 in. 2 A r 2 ( ) 2 in. 2 P 18 in. The circumference is multiplied by 1 2. The area is multiplied by ( ) 2 or. Effects of Changing Area A square has side length 10 yd. If the area is quadrupled, what happens to the side length? The area of the original square is A s 2 2 yd 2. If the area is quadrupled, the new area is yd 2. Set the new area equal to s 2. Take the square root of both sides. Simplify. The side length is multiplied by. s 2 s s 20 yd 134 Holt Geometry
9B Ready To Go On? Problem Solving Intervention 9-5 Effects of Changing Dimension Proportionally When all of the dimensions of a figure are changed proportionally, the new figure will be similar to the original figure. Erin has a vegetable garden that is 5 ft long and 7 ft wide. It costs $3.49 to fertilize her garden. She wants to enlarge the garden by making each side length 3.5 times as long. How will the area of the new garden compare to the area of the old garden? How much will it cost to fertilize her new garden? Understand the Problem 1. What two questions must you answer? 2. Will Erin s new garden cost more or less to fertilize than her old garden? Make a Plan 3. What is the area of the original garden? 4. What are the dimensions of the new garden? 5. What is the area of the new garden? Solve 6. How does the area of the new garden compare to the area of the old garden? 7. How much will it cost to fertilize the new garden? Look Back 8. Divide the area of the new garden by the area of the old garden. 9. 3. 5 2 ; Does your answer in Exercise 8 match this answer? 10. Explain why these numbers should match. 135 Holt Geometry
9B Ready To Go On? Skills Intervention 9-6 Geometric Probability Find this vocabulary word in Lesson 9-6 and the Multilingual Glossary. Vocabulary geometric probability Using Angle Measures to Find Geometric Probability Use the spinner to find the probability of each event. A. the pointer landing on the dotted region The angle measure in the dotted region is. The angle measure of the circle is. The probability of the pointer landing on the dotted region is P angle measure 360 130 60 80 90 360 Substitute the angle measure and simplify. B. the pointer landing on gray or white The angle measure in the gray region is. The angle measure in the white region is. Add the angle measures in the gray and white regions. The probability of the pointer landing on gray or white is P angle measure 360 360 Substitute the sum of the angle measures and simplify. C. the pointer not landing on the checked region The angle measure of the checked region is. Subtract this angle measure of the checked region from 360. 360 The probability of the pointer not landing on the checked region is P angle measure 360 360 Substitute the difference of the angle measures and simplify. 136 Holt Geometry
9B Ready To Go On? Problem Solving Intervention 9-6 Geometric Probability The probability of an event can sometimes be modeled using a line segment. The programming on a local morning radio show has the following cycle: news and traffic for four minutes, music for nine minutes, and commercials for two minutes. If you turn on the radio at a random time, find the following: a. the probability that a commercial will be playing. b. the probability that music will not be playing. Understand the Problem 1. How can you use geometry to represent this situation? 2. The question asks you to find a probability. What kind of number will your answer be? Make a Plan 3. Draw and label a segment to represent the number of minutes that news and traffic, music, and commercials are on air. 4. What is the total number of minutes in a cycle? news and traffic music commercials A 4 B 9 C 2 D Solve 5. Find the probability that a commercial will be playing. P number of minutes of commercial number of minutes in cycle 6. For how many minutes in a cycle is music not playing? 7. Find the probability that music will not be playing. P number of minutes music does not play number of minutes in cycle Look Back 8. Convert your answer in Exercise 5 to a percent. Is it reasonable that commercials play approximately 13% of the time? 9. Convert your answer in Exercise 7 to a percent. Is it reasonable that music does not play 40% of the time? 137 Holt Geometry
9B Ready To Go On? Quiz 9-4 Perimeter and Area in the Coordinate Plane Draw and classify the polygon with the given vertices. Find the perimeter and area of the polygon. 1. A( 1, 4), B(2, 4), C(2, 3), D( 1, 3) 2. E(6, 2), F(3, 3), G( 1, 3), H( 4, 2) y 2 2 2 2 4 x 2 4 4 y 2 2 4 6 2 x Find the area of each polygon with the given vertices. 3. J(4, 5), K( 2, 1), L(1, 3), M(6, 2) 4. N( 3, 2), M(1, 5), P(5, 1), Q(1, 5) 6 4 y 4 2 y 2 2 2 2 4 6 4 x 4 2 2 4 6 2 4 6 x 9-5 Effects of Changing Dimensions Proportionally Describe the effect of each change on the perimeter and area of the given figure. 5. The side length of the square is doubled. 9 ft 6. The diagonals of a rhombus in which d 1 8 m and d 2 7 m are both multiplied by 1 2. 138 Holt Geometry
9B Ready To Go On? Quiz continued 7. The base and height of the rectangle are both tripled. 14 in. 6 in. 8. The base and the height of a right triangle with base 12 cm and height 18 cm are multiplied by 1 3. 9. A square has vertices ( 4, 1), (1, 1), (1, 4), and ( 4, 4). If you multiply the area by 9, what happens to the side length? 10. A sports photographer sells team pictures. The cost of each picture is based on its area. Wallet-sized pictures measure 4 cm by 6 cm and cost $1.39 each. The photographer sells a larger picture that is three times the length and three times the width of the wallet-sized picture. How much does the larger picture cost? 9-4 Geometric Probability Use the spinner to find the probability of each event. 11. the pointer landing in the gray or white region 12. the pointer landing in the striped region 13. the pointer not landing in the dotted region 40 125 105 90 14. the pointer landing in the dotted or white region 15. A television station plays 14 commercials every half hour. Each commercial is 30 seconds long. If you turn on the TV at a random time, find the probability that a commercial will be playing. 139 Holt Geometry
LESSON 9B Ready To Go On? Enrichment Applying Geometric Formulas 1. The radius of the circle is 9 yd. Find the probability that a point chosen randomly inside the circle is in the shaded area. Round to the nearest hundredth. 2. Find the probability that a point chosen randomly inside the hexagon is in the shaded area. 5 in. 3. Find the probability that a point chosen randomly inside the circle is in the shaded area. 3 m 150 A 4. Estimate the shaded area inside the figure. Round to the nearest hundredth. 4 2 y 6 4 2 2 2 4 6 4 x y 5. Find the area and perimeter of the octagon. 2 2 2 2 x 140 Holt Geometry