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Name_____________________________ Period_______ Concepts of Geometry Homework Packet Page 1 Medians, Altitudes and Angle Bisectors Part One. Draw each of the following. Make sure all parts are labeled and marked appropriately. 1. PS is a median of PQR and 2. QT is an angle bisector of PQR S is between Q and R and T is between P and R 3. AC and BC are both altitudes of ABC 4. PT and RS are medians of PQR and intersect at point V Part Two. Tell if each of the following statements are ALWAYS, NEVER, or SOMETIMERS true. 5. The three medians of a triangle intersect at a point inside the triangle. 6. The three angle bisectors of a triangle intersect at a point outside the triangle. 7. The three altitudes of a triangle intersect at a vertex of the triangle. 8. If a median of a triangle is also an altitude, then the triangle is isosceles 9. A point on the angle bisector of a triangle is equal distances from the sides of the triangle. Page 2 Page 3 Advanced Parts of a Triangle For questions #1-4, use a triangle with the coordinates of R(3, 3) S(-1, 6) and T(1, 8) 1. If RX is a median of this triangle, what are the coordinates of X? 2. What is the length of segment RX? 3. What is the slope of RX? 4. If SP is an altitude of the triangle, what is the slope of SP? Use this information for questions #5-7: YV is an angle bisector of XYZ and ZYX = 8x – 6 and XYV = 2x + 7. 5. What is the value of x? 6. What is the measure of ZYX? 7. What is the measure of ZYV? 8. Find the values of x and y if AD is an altitude and a median of the triangle. . 4x - 6 10y - 7 5y + 3 Page 4 Solving Triangles Solve for the value of x and y in each figure if ABC DEF 1. C F 4y 28 18 E D 2x + 12 A B 20 2. B 44 2x - 20 40 D E A C 4y + 12 36 F A 3. E 65 45 5x + 35 10y D F 50 B C Page 5 Triangle Sides and Angles Part One. For #1-2, list the angles of each triangle in order from greatest to least. 1. 2. M A 7.2 11 L 9 8 7.8 C B 8 N Part Two. For #3-4, list the sides in order from shortest to longest. 3. 4. R F 45 30 Q 80 55 89 61 H G S Part Three. For #5-6, use the given information about a PQR to place the sides in order from longest to shortest. 5. P = 7x + 8, Q = 8x – 10 and R = 7x + 6 6. P = 3x + 44, Q = 68 – 3x and R = x + 61 Page 6 Triangle Inequalities Part One. Determine whether each set of numbers could be the sides of a triangle. Write YES or NO. 1. 12, 11, 17 2. 1, 2, 3 3. 4, 9, 4 4. 2.5, 6, 6.5 5. 12, 2, 14 6. 2, 12, 12 7. 9, 40, 41 8. 5, 100, 101 9. 204, 7, 215 Part Two. Two sides of a triangle are 18 and 21. Tell if each of the following numbers could be the third side of that triangle. Write YES or NO. 10. 10 11. 40 12. 7 13. 21 14. 3 15. 57 Part Three. The measures of two sides of a triangle are given. Between what two numbers must the third side be? 16. 12 and 15 17. 4 and 13 18. 21 and 17 Page 7 Page 8 GUESS AND CHECK (Part one) 1. Sum of two numbers is 15 Product of those numbers is 56. Find the two numbers. 2. Sum of two numbers is thirteen. Difference of those numbers is 3. Find those two numbers. 3. Sum of two numbers is 18. Quotient of those numbers is 2. Find those two numbers. 4. Sum of two numbers is 32. Product of those numbers is 231. Find those two numbers 5. Sum of three numbers is 17. Product of those numbers is 168. Find all three numbers. Page 9 GUESS AND CHECK (Part two) 1. On a farm, there are chickens and rabbits. All together, there are seven heads, but 20 legs. How many of each type of animal was on the farm? 2. A teacher tells his class that their next test is worth 100 points and has 30 problems on it. Some of the problems are open ended which are worth 6 points while others are multiple choice and only worth two points. How many of each type of question are on this test? 3. John has quarters and dimes in his pocket. When counted together, he found he has $1.05 with just six coins. How many of each coin does he have in his pocket? 4. Tracy’s three math test grades had an average of 80. Her first test was ten points more than her second. The second was test was twenty points less that the third. What were her three test grades? 5. Find three consecutive odd numbers whose sum is 39. Page 10 TOO MUCH INFORMATION 1. Gracy High School ordered 17 computers and 9 dvd players. The computers sold for $675 each, but the school got a discount and only paid $558 each. Each dvd player cost $65 each. How much did the school pay total for the computers? 2. Harry works 17 hours a week and makes $9.50 an hour. He also spends 10 hours a week practicing his trumpet. He saved all his earning for seven weeks and purchased a new trumpet for $675. After this purchase, how much of his savings did Harry have left? 3. The fastest speed that Mona can get her car to go is 80 miles per hour. However, Candy can get her car to 92 miles per hour and Janice gets her car up to 98 miles per hour. If these women drive their cars at their fastest rates possible, then how much further does Janice travel in a half hour than Mona does? 4. Lily bought 6 hamburgers at 79 cents each, nine French fries at 72 cents each, six sodas at $1.09 each, and two milkshakes for $2.15 each. How much did Lily spend total on the drinks? Page 11 WORKING BACKWARDS 1. Susan made a deposit of $47 into her checking account. Then, she wrote a check for $112. After that, her balance was $1102. How much was in her account originally before the deposit and check cashing? 2. Ace gave his friend Benny 12 of his marbles. After that, he had 16 marbles left. How many marbles did Ace originally have? 3. Begin with a number. Add seven to that number. Multiply that result by three. Then, subtract ten from that result. The final number is 62. Find the original number. 4. Gary told his niece that is you double his age and then add three, you get 87. How old is Gary? 5. A man got into an elevator on a specific floor. He rose two floors, then descended three floors, then rose up again four floors. If he then got off on the 7th floor, on what floor did he begin? Page 12 MAKE A LIST 1. A DJ has four different songs to play. He hasn’t decided the order to play them in yet, though. How many different choices does he have? 2. A vending machine will take nickels, dimes or quarters. Each item in the machine costs 65 cents. How many different combinations of coins could be used to purchase an item? 3. A student must answer any four of five questions on a history exam. How many different choices does that student have to answer four of five questions? 4. Sarah bought 2 skirts, four blouses, and 3 pairs of shoes. How many different combinations of outfits does that give her? Page 13 MAKING A TABLE 1. Joey placed $400 into a savings account. Each year, 5% interest is gained on the amount in the account. Complete this table if Joey does not withdrawal any money: Year 0 1 2 3 4 Amount $400 2. Mitchell wrote a book and had it published. The first month it was released, he sold 5 copies. The second month, he sold 15 copies. In the third month, there were 35 sales and in the fourth, there were 65. Complete this table to continue such a pattern: Month 4th 5th 6th 7th 8th Copies 65 3. Zeus had $600 in his savings account, but withdrew $25 each month. Kara started with $340, but deposited $70 each month. Complete this table. Month 0 1 2 3 4 Zeus $ $600 Kara $ $340 4. Betty sells baked goods at a local market stand. She currently charges $1.20; however, she wants to raise the price to $2.10. She does not want to do it all at once, though. She will raise the price by the same amount each week until the price reaches $2.10. Complete this table based upon such information: Week 0 1 2 3 4 5 Price $1.20 $2.10 Page 14 DRAW A PICTURE 1. Rose has a garden which is 10 feet long and six feet wide. She wants to place a fence around her garden. To do so, she first places posts in the ground. She begins in one corner, and places a post every 2 feet apart. How many posts will she need? 2. Amber flew a kite the highest of all her friends. Bonnie’s kite was 50 feet lower than Amber’s but 100 feet higher than Carla’s. Carla’s kite was 100 feet higher than Derek’s. Derek’s kite was 120 feet above the ground. How high was Amber’s kite? 3. A round circle has three straight line cuts made into it. What is the largest number of pieces that it could be cut into? 4. The deck in Mary’s yard is in the shape of a circle. The circle has 6 posts dispersed around its circumference. If you want to attach one string from each post to every other post, what is the least number of strings that will be needed? 5. Gary started at a point and walked 20 feet north. He then went 40 feet east, 20 feet south, and 10 feet west. How far is Gary from his starting point? Page 15 Page 16 USING THE HSPA REFERENCE SHEET 1. How many ounces are in three pounds? 2. What fraction is equal to pi ( )? 3. Find the volume of a rectangular prism when its length is 3, width is 5 and height is 6. 4. Find the surface area of the same prism from question #3. 5. How many seconds are in one day? 6. Which is longer: four feet, one yard OR forty inches? 7. How many degrees are in a circle? 8. Find the area of a circle when its radius is 5. 9. How many fluid ounces are in one quart? 10. A rectangle has a perimeter of 40 and a width of 6. Find its length. Page 17 HSPA Multiple Choice questions (Circle your answers) 1. Hank receives an allowance each week from his parents. He is required to save 10% of it in a bank account. What would Hank’s minimum allowance be so that he would have $15 to actually spend? A. $13.50 B. $15.10 C. $16.50 D. $16.67 2. In the five basketball games of the Holiday Tournament, four of the scores for Joe’s team were 109, 105, 97 and 92. If the mode of all five scores was 92, what was the mean of those five scores? A. 99 B. 97 C. 95 D. 93 3. Jason’s older brother owes him $200. He offers to pay Jason $120 today and then tomorrow to pay him ¼ of what he paid him today. If this pattern of paying ¼ of what he paid the day before continues, how much will Jason’s brother have paid by the end of the third day? A. $160.00 B. $157.50 C. $155.00 D. $149.75 4. Jan is building a scale model of a house. If the actual house is 86 feet wide and 172 feet long, what will be the length in inches of the model if it is 18 inches wide? A. 3 B. 6 C. 12 D. 36 5. Alex jogged 8 miles north, then turned due west and jogged 6 more miles. What is the shortest distance from Alex’s current place to his starting point? A. 9 B. 10 C. 14 D. 24 6. Laura wanted to raise her science grade from 80 to 84. What percent increase would that represent? A. 4% B. 5% C. 6% D. 7% 7. The Garden Center Manager, Mr. Stern, ordered two sizes of grass seed bags for the spring sale. He ordered three times as many of the 25-pound bags as he did of the 40- pound bags. The total weight of all of the seed was 920 pounds. How many smaller bags of seed did Mr. Stern order? A. 6 B. 18 C. 24 D. 81 Page 18 Part Eight: Quadrilaterals Page 19 Parallelograms Part One. Use parallelogram ABCD to complete the blanks in each statement. 1. AB is parallel to _____ A B 2. DA _____ 3. ADC _____ E 4. CDA _____ D C 5. DE is congruent to _____ 6. BAC _____ Part Two. Use parallelogram EFGH to tell if each statement is true or false. 7. FE || GH 8. FDE HDG F 9. FGH FEH G 10. FD is congruent to DG D 11. FHE GHE E 12. DE = ½EG H Page 20 Advanced Parallelograms 1. If quadrilateral SLAM is a parallelogram and S = 92, then find the measures of the other three angles. 2. If DUNK is a parallelogram, DU = 3x + 6, UN = 8y – 4, KN = 8x – 4 and KD = 2y + 14, then find the perimeter of DUNK. 3. Given parallelogram PQRS with P = y and Q = 4y + 20, find the measure of R 4. In parallelogram ABCD, if A = x + 75 and B = 3x – 199, then find the measure of D. 5. Find the values of x and y: 6y 30 5x + 10 For #6-8, draw a parallelogram ABCD with diagonals intersecting at point T. Use that figure to answer these questions. 6. If ABC = 137, then find DAB 7. If AC = 5x – 12 and AT = 14, then find the value of x. 8. If BC = 4x + 7 and AD = 8x – 5, then find the value of x. Page 21 Rectangles Use rectangle JKLM to answer questions #1-2 to find the value of x: M 1. LP = 3x + 7 and MK = 26 L 2. LJ = 4x – 12 and KM = 7x – 36 P K J Use rectangle MATH to answer questions #3-8: 3. If MP = 6, then find HA 4. If MH = 8, then find AT M A P 5. If HP = 3x and PT = 18, then find x H T 6. If MPH = 55, then find MHP 7. If MPA = 110, then find APT 8. If MA = 2x + 4, AT = 8, TH = 3y + 5, and MH = 2y – 2, then find the values of x and y. Page 22 Squares For #1-5, draw a square ABCD with diagonals intersecting at point E. Use that figure to answer the questions. 1. If AEB = 3x, then find the value of x. 2. If BAC = 9x, then find the value of x 3. If AB = 2x + 4 and CD = 3x – 5, then find the length of BC 4. If DAC = y and BAC = 3x, then find the values of x and y. 5. If AB = x + 15 and BC = 2x, then find the length of CD. 6. If WXYZ is a square, then what is the measure of ZXY? Page 23 Rhombus #1-6: Use rhombus BEAC and the fact that BA has a length of 26 to tell if each statement is TRUE or FALSE. 1. CE = 26 2. HA = 13 B E 3. BA EC H 4. BHE AHC A C 5. BEH EBH 6. CBE and BCA are supplementary. For #7-10, use rhombus SRQP. 7. If ST = 13, then find SQ S R 8. If PRS = 17, then find QRS T 9. What is the measure of STR? P Q 10. If SP = 4x – 3 and PQ = 18 + x, then find the value of x. Page 24 Quadrilateral Chart Complete each box with a “YES” or “NO”. Property Parallelogram Rectangle Square Rhombus Trapezoid Opposite sides are equal Diagonals are perpendicular All four sides are equal Opposite sides are parallel Diagonals bisect each other Diagonals are congruent Only one pair of parallel sides Diagonals bisect opposite angles Page 25 Trapezoid For #1-2, solve for x. (Assume the line inside the trapezoid is a median) 3x + 5 1. 27 2. x 55 13 7x - 10 For #3-6, use trapezoid PQRS as seen to the right. TV is its median. 3. IF PS = 20 and QR = 14, then find the length of TV P T 4. If QR = 14.3 and TV = 23.3, then Q find the length of PS 5. If TV = x + 7 and PS + QR = 5x + 2, R then find the value of x V S 6. If RVT = 57, then find QTV. 7. UR is the median of a trapezoid with bases ON and TS. If the coordinates of the points are U(1, 4) R(8, 4) O(0, 0) and N(8, 0), then find the coordinates of points T and S. Page 26 Page 27 Proportions Tell if each set of fractions forms a proportion. Write “YES” or “NO” 6 22 4 16 4 12 8 16 1. 2. 3. 4. 8 28 5 20 11 33 9 17 Solve each proportion for the value of x. (Note: some answers may be decimals) 3 x x 3 x 7 5. 6. 7. 4 8 45 15 9 16 x .21 858 702 x 11.75 8. 9. 10. 8 2 x 900 33 35.25 x 3 3 1 3 12 11. 12. 13. 4 2 x 6 1 x Substitution. If a = 3, b = 2, c = 6, and d = 4 , then do each of the following form a proportion? Write “YES” or “NO” b d a b c d 14. 15. 16. a c c d b a ab cd d c ab cd 17. 18. 19. b d b a c b Page 28 Advanced Proportions Solve each proportion for the value of x. Some answers will be decimals. 3 x2 x 2 14 1. 2. 5 6 5 10 5 x 14 x 9 x 10 3. 4. x 3 10 10 11 x 7 x 12 x2 5 5. 6. 4 6 7 7 x 1 1 1 2 7. 8. 3 5 3 x2 x 2 x 1 5 x 13 9. 10. 3 6 3 x 12 Page 29 Proportion Word Problems Solve each word problem by using proportions. 1. A 96-mile trip requires 6 gallons of gas. At that rate, how many gallons would be needed for a 152 mile trip? 2. When a pair of blue jeans is made, the leftover denim scraps can be recycled to make pencils. One pound of denim is left over after making every five pairs of jeans. How many pounds are left over after 250 jeans are made? 3. In the first 30 minutes of the opening day of the Texas State Fair, 1252 people entered through the gates. If this rate continued, how many people would enter during the operating hours of 8am to 11pm? 4. Josh finished 24 math problems in one hour. At that rate, how many hours will it take him to complete 72 problems? 5. To make a model of the Guadelupe River bed, Harold used 1 inch of clay for every 5 miles of actual river length. His model was 50 inches long. How long is the actual river? 6. A recipe calls for 5 pounds of cocoa beans for every 4 gallons of milk. How much milk is needed if 75 pounds of cocoa beans is used? Page 30 Similar Figures (Figures in this lesson are not drawn to scale) Use the similar quadrilaterals to the right for questions #1-3: (Note: QRSP BCDA) Q R 1. If AB = 4, AD = 8 and PQ = 6, find PS P S 2. IF RS = 4.5, CD = 6.3 and BC = 7, find QR. B C 3. If CD = 22, DA = x - 2 and SR = 33, A find SP. D Use the trapezoids to the right to answer questions #4-8. Note that TFEL KROC. 4. How long is KR? 15 T F 12 12 5. How long is RO? K R 6. How long is CO? L E 6 8 C O 7. What is the perimeter of TFEL? 8. What is the perimeter of KROC? Use the quadrilaterals to the right to answer questions #9-11. Note that BCDA QRSP. 9. What is the value of x? B x+5 Q x+3 5 C R 10. How long is BC? 4 A P S 11. How long is QR? D Page 31 Figure Proportions For #1-4, tell if each proportion is TRUE or FALSE. C BC AB CB CA 1. 2. ED CD CD CE B D AB DE BA CA 3. 4. BC DC DE CE E A For #5-10, fill in the blanks with the correct segment. D YO AE YB ? 5. 6. A OB ? OB ER Y ? YB DY DA E 7. 8. AE YO YO ? O R DR DB ? DO 9. 10. ? YB AE YO B For #11-14, tell if each proportion is TRUE or FALSE. T L A TM TL BA ML 11. 12. MB LA MB LA TB TA TM TB M 13. 14. MB LA ML BA B Page 32 Advanced Figure Proportions Solve for all missing values (x and y) 8 6 1. 7 x 2. x+5 x 10 12 2y + 6 2x x + 12 3. 4. y+8 5x - 3 9 2y - 4 x 24 18 5. 6. 4 5 6 x-3 x 7 x+3 x+3 Page 33 REVIEW 1. Use a triangle with the coordinates of R(3, 5) S(-1, 10) and T(1, 4) (a) If RX is a median of this triangle, what are the coordinates of X? (b) What is the length of segment RX? (c) What is the slope of RX? (d) If SP is an altitude of the triangle, what is the slope of SP? 2. Solve for the value of x and y in these figures if ABC DEF C F 4y 36 16 E D 2x + 12 A B 20 3. Use the given information about a PQR to place the sides in order from longest to shortest. P = 2x + 8, Q = 5x – 5 and R = 3x + 7 4. Two sides of a triangle are 15 and 21. Tell if each of the following numbers could be the third side of that triangle. Write YES or NO. (a) 36 (b) 40 (c) 7 5. The measures of two sides of a triangle are given. Between what two numbers must the third side be? 12 and 19 Page 34 6. Solve each word problem: (a) Sum of two numbers is 18 and the product of those numbers is 80. Find the two numbers. (b) On a farm, there are chickens and rabbits. All together, there are nine heads, but 38 legs. How many of each type of animal was on the farm? (c) Begin with a number. Add seven to that number. Multiply that result by two. Then, subtract ten from that result. The final number is 24. Find the original number. (d) A vending machine will take nickels, dimes or quarters. Each item in the machine costs 75 cents. How many different combinations of coins could be used to purchase an item? (e) Zeus had $800 in his savings account, but withdrew $25 each month. Kara started with $450, but deposited $60 each month. Complete this table. Month 0 1 2 3 4 Zeus $ $800 Kara $ $450 (f) Gary started at a point and walked 25 feet north. He then went 47 feet east, 25 feet south, and 10 feet west. How far is Gary from his starting point? (g) How many seconds are in one day? (h) Jason’s older brother owes him $260. He offers to pay Jason $120 today and then tomorrow to pay him ¼ of what he paid him today. If this pattern of paying ¼ of what he paid the day before continues, how much will Jason’s brother have paid by the end of the third day? Page 35 7. If quadrilateral SLAM is a parallelogram and S = 91, then find the measures of the other three angles 8. Use rectangle JKLM to answer questions to find the value of x: (a) LP = 3x + 7 and MK = 26 M L P (b) LJ = 4x – 12 and KM = 6x – 36 K J 9. Draw a square ABCD with diagonals intersecting at point E. Use that figure to answer the questions. (a) If AEB = 5x, then find the value of x. (b) If BAC = 9x, then find the value of x (c) If AB = 2x + 4 and CD = 3x – 9, then find the length of BC 10. For #3-6, use trapezoid PQRS as seen to the right. TV is its median. (a) IF PS = 20 and QR = 16, then find the length of TV P (b) If QR = 14.3 and TV = 25.3, then T Q find the length of PS (c) If TV = x + 7 and PS + QR = 5x + 2, R then find the value of x V S (d) If RVT = 59, then find QTV. Page 36 For #11-14, solve each proportion. 4 x2 x 2 14 11. 12. 5 10 5 10 x5 7 x 8 x 10 13. 14. x 3 5 5 4 15. A 96-mile trip requires 8 gallons of gas. At that rate, how many gallons would be needed for a 152 mile trip? 16. Use the similar quadrilaterals to the right for questions : (Note: QRSP BCDA) Q R (a) If AB = 4, AD = 5 and PQ = 8, find PS P S (b) IF RS = 4, CD = 3 and BC = 6, find QR. B C (c) If CD = 22, DA = 32 and SR = 33, A find SP. D 17-18: Solve for x .17 18. 4 6 8 x-3 x 8 x+3 x+4 Page 37

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