Mathematics. Guide GEO Extended answer Problem solving GEO.02 Extended answer Problem solving

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1 1- Contents Mathematics Guide Question Item Objective Type Skill GEO Multiple-choice answer Applications GEO Extended answer Problem solving GEO.02 Extended answer Problem solving GEO Multiple-choice answer Applications GEO.02 Extended answer Problem solving GEO.02 Extended answer Problem solving GEO.02 Extended answer Problem solving GEO.02 Extended answer Problem solving GEO.02 Extended answer Problem solving GEO.02 Extended answer Problem solving GEO.02 Extended answer Problem solving GEO.02 Extended answer Problem solving GEO.02 Extended answer Problem solving GEO.02 Extended answer Applications GEO Extended answer Problem solving GEO.02 Extended answer Applications GEO.02 Extended answer Problem solving GEO.02 Extended answer Concepts GEO.02 Extended answer Problem solving GEO.02 Extended answer Problem solving

2 GEO.02 Extended answer Problem solving 2- Correction key A

3 2 Example of an appropriate solution Find the measure of BON: m MOA = m BON = = 13 2 Find the measure of MAO: m MAO = 180 ( ) = 107 Find the measure of segment AO: sin ( MAO) sin ( AMO) m MO ( 107 ) sin ( 60 ) sin 100 cm = = m AO = m AO m AO sin m AO ( 60 ) sin ( 107 ) cm m AO cm Find the measure of OAB: m OAB = = 73 Find the measure of ABO: 100 cm m ABO = 180 ( ) = 43 Find the measure of segment AB:

4 Answer: Note: sin ( ABO) sin ( AOB) m AO = ( 43 ) sin ( 64 ) sin = cm m AB = m AB m AB sin m AB ( 64 ) sin ( 43 ) cm m AB cm cm From Point A to Point B, the marble travelled cm Accept answers in the range [119, 120]. A student can also approach this solution by finding the measure of segment BO instead of segment AO. A student can also approach this solution by finding 200 ( m MA + m BN). Students who have found the measure of either segment AO or segment BO have shown they have a partial understanding of the problem.

5 3 Example of an appropriate method Measure of CBD 180 ( ) = 60 Length of segment BD m BD sin 35 = 5.2 sin 60 m BD = metres Length of segment AB m AB = cos m AB = metres Length of segment BE Answer: Note: m BE = 2.5(3.326) = To the nearest tenth of a metre, pole BE measures 8.3 metres. The student who determined the length of BD, 3.44 metres, has demonstrated a partial understanding of the problem.

6 4 A

7 5 Example of an appropriate method SRL = 55 (corresponding angles, parallel lines) In small SLR 1030 m sin 55 = m SR m SR sin 55 = 1030 m 1030 m m SR = sin 55 m SR = m In the large SMB cos 55 = 980 m m SB m SB cos 55 = 980 m 980 m m SB = cos 55 m SB = m m RB = ( ) m m RB = 451

8 6 Answer: Because there are many ways to do this problem, and answers will vary as a result of rounding, accept 450 m to 453 m. Example of an appropriate method Order lengths of fencing and divide into quintiles: Using Hero's formula to find area of triangle: Answer: A = 34( 34 20)( 34 21)( 34 27) = = = Rupert will enclose m 2 of land. Note: Do not deduct any marks for incorrect rounding.

9 7 Example of an appropriate solution Matthew's route Distance from town A to cabin x cos 25 = 125 x Distance from cabin to town B x sin 25 = 125 x Total distance Difference in distances Answer: = km = km Cynthia's route Measure of angle A = sin A sin 120 sin A m A Measure of angle B 180 ( ) = Distance from town A to restaurant 125 b = sin 120 sin b Total distance = km To the nearest tenth, the difference between the two routes is 23.9 km.

10 Note: Accept answers in the range of 23.8 km to 24 km.

11 8 Example of an appropriate method Length of segment AD cos 20 = m AD 7.5 m AD = Measure of angle ABD m ABD = 180 m BAD m BDA m ABD = m ABD = 130 Length of segment BD m BD = sin BAD m BD = sin 20 m AD sin ABD sin 130 m BD Answer: Rounded to the nearest tenth, the length of the beam represented by segment BD is 3.1 m. Note: Students who use an appropriate method in order to determine the length of segment AD have shown that they have a partial understanding of the problem.

12 Do not penalize students who did not round off their final answer or who made a mistake in rounding it off.

13 9 Example of an appropriate solution Measure of side CD cos 26 = m CD 18 m CD Area of triangle ADC A Measure of angle B b h = m B = 180 ( ) = 68 Measure of segment AB 18 = sin 68 m AB sin 34 m AB Measure of segment BC m BC = sin sin 68 m BC C D 18 m 7.89 m A 78 B

14 Area of triangle ABC Perimeter = perimeter = p 2 A = p (p a) (p b) (p c) A ( )( )( ) A Total area of plot Answer: m 2 Helen will need to buy 160 pieces of sods of grass.

15 10 Example of an appropriate method Measure of segment AC tan 65 = 15 m AC 15 m AC = tan 65 m AC = Measure of segment BC m BC 7 4 = 3 Angle of elevation of point B 15 tan B = 3 tan B = 5 m B = Answer: The angle of elevation of the observer located at B is 79. A 65? 4 m B D C 15 m

16 11 Example of an appropriate method Height h 1 Height h 2 h1 tan 40 = 15 h 1 = 15 tan 40 h h2 tan 28 = 15 h 2 = 15 tan 28 h Height of the building h 1 + h Answer: The height of the building is m m 40 h 2 h 1

17 12 Example of an appropriate method Area of rectangle A = L l A = A = 5000 Area of triangle Total area Heron's formula: p(p a)(p b)(p c) A = Asking price A (117 50)(117 72)( ) A t = Area of rectangle + area of triangle A t = Price A t 1.35 = Answer: To purchase this property, Julio must pay $ Accept any answer in the interval [$ , $ ] also.

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19 13 Example of an appropriate method of solution Measure of BC 64.3 m BC = sin 100 sin 30 m BC Perimeter of triangle ABC = s = semi-perimeter s Area of triangle ABC A 73.47( )( )( ) (Heron's formula) Answer Rounded to the nearest m 2, the area of the plot of land is 804 m 2. Accept an answer in the interval [803, 804].

20 Alternate solution: By drawing an altitude BD from B to AC, as shown in the diagram, and using the 1 fact that it is opposite a 30 angle, we find m BD to be (50) = 25 m. 2 Using the formula for the area of a triangle 2 1 Base Altitude we obtain 2 1 (25)(64.3) = Area of plot of land to the nearest m 2 is 804 m 2 A 50 m B 64.3 m D C

21 14 Example of an appropriate algebraic solution Krystal's distance from the base of the house tan 20 = m AB = 4 m AB 4 tan 20 Height of Krystal's house tan 40 = m BD m m BD = tan m Distance from window to the roof of the house m BD m BC = = m Answer The distance from window to the roof of the house is 5.2 m.

22 15 Example of an appropriate algebraic solution Volume of smaller box l w h = = 64 cm 3 Ratio of volumes 1728 = 64 Ratio of edges = Height of larger box Height = 8 3 = 24 cm Answer The height of the larger box is 24 cm.

23 16 Work : (example) Measure of BC cos 53 m BC Measure of BD Result sin 53 m BD m BD = = = = m BC cos 53 m BD m BC cos53 sin 53 The measure of segment BD is about 7.2 cm.

24 17 Example of an appropriate solution Measure of BD and of angle ADB m BD = m CD = 5 cm m ADB = 180 ( ) = 95 Measure of AD Measure of AB 5 m AD = sin 50 sin 35 5 sin 35 m AD = 3.74 sin 50 5 m AB = sin 50 sin 95 5 sin 95 m AB = 6.5 sin 50 Measure of half the perimeter (p) p 2 1 ( ) = 7.62 Area of triangle ABD S = p(p a) (p b) (p c) A 50 C D 5 cm B

25 18 S 7.62(7.62 5) ( ) ( ) S = Answer The area of triangle ABD is 9.31 cm 2. Accept answers that fall within the interval [9.21, 9.32]. Example of an appropriate solution I would find : The area of the triangle using Heron s formula Area = b h 2 Solve for h, h = 2 area b Any other plausible answer is acceptable.

26 19 Example of an appropriate solution Example 1 Measure of DC : Sin A = m DC m AC m DC = m AC sin A or m DC = 1 m sin m DC m Measure of ABC sin B = m DC m BC m B sin B = Final answer : Angle ABC measures or Example 3 The area of ABC using Heron s formula Example m 1m = sin sin ABC 1 sin sin πabc = 1.5 sin ABC m m ABC Final answer A D 46.57E C Angle ABC measures B

27 P = 2 1 ( ) = 2.25 Area = 2.25(2.25 1) (2.25 2) ( ) Area m 2 Measure of DC Area = Height = B h 2 2 area base Height = Measure of B sin B = m DC m BC m B sin B = Final answer Angle ABC measures

28 20 Example of an appropriate method Height of the tree tan 40 = m OP 10 m OP 8.39 Distance between Anthony and the tree tan m AP m AP Distance between Anthony and Steven Answer m AE m AE The distance between Anthony and Steven is metres. Accept all result in [13, 13.1] A S P 10 m O

29 21 Example of an appropriate method Length of side a sin 40 = 6 a Length of side b a = 6 sin 40 3,86 sin 50 = 6 b b = 6 sin 50 4,60 a b Area of triangle: = Quantity of paint Answer 1 L covers 2 m = (4 litres is not enough; she must buy 5 litres) Mara will have to buy 5 litres of paint. 50 a 6 m b 40

30 Mathematics Question Booklet Marie-Pierre is standing exactly half way between two office-building towers. These towers are on opposite sides of the street. Using a clinometer (a device used to measure angles) she calculates the angle formed by the top of each building. The two readings are 72 and 40. The distance between the two buildings is 66 metres. What is the difference between the heights of the two buildings? Round your answer to the nearest tenth of a metre m A) 73.9 m C) m B) m D) m

31 2 In a science experiment, a class wanted to calculate the distance a marble travelled from point A to point B on a slope with the measurements shown below: 100 cm M O A cm To help her students, the teacher gave out additional information: m AMO = 60 m AOB = 64 MOA BON How many centimetres did the marble travel from point A to point B? Show all your work. B N

32 3 A drawing of a banner is shown in the diagram on the right. Pole BE, supporting the banner, is 2.5 times the length of segment AB. What is the length of pole BE to the nearest tenth of a metre? Show all your work B A E D m C

33 4 Marie-Pierre is standing exactly half way between two office-building towers. These towers are on opposite sides of the street. Using a clinometer (a device used to measure angles) she calculates the angle formed by the top of each building. The two readings are 82.1 and The distance between the two buildings is 66 metres. What is the shortest distance between the two rooftops, represented by AB? Round your answer to the nearest tenth of a metre. A) m C) m B) m D) m

34 5 A street map of a part of a small town is shown below m (library) L (museum) M S (school) 980 m R (restaurant) 55 B (beach) What is the shortest distance from the beach to the restaurant? Round your answer to the nearest metre. - Two of the streets are parallel. - The distance from the museum to the beach is 980 m. - The distance from the school to the library is 1030 m.

35 7 Cynthia and Matthew each rode from town A to town B on their bicycles. Their routes are represented in the diagram on the right. The diagram is not drawn to scale. The direct route between the two towns is 125 km. Restaurant Cynthia's route Town A Cynthia's route km To the nearest tenth, how much longer is Matthew's route than Cynthia's? Show all your work. 125 km Town B Matthew's route Matthew's route Cabin

36 8 9 The structure represented in the diagram below is used in a warehouse to lift objects off the ground and onto a shelf. A 7.5 m B E? C D Shelf Rounded to the nearest tenth, what is the length of the beam represented by segment BD? Show all your work. Helen's plot of land is represented by the diagram on the right. She wants to cover it with pieces of grass sod that each measure 1 square metre. How many pieces of grass sod will Helen need to buy? Show all your work. C D 18 m 7.89 m A 78 B

37 10 11 Two bird watchers, 4 metres apart, are located at positions A and B as shown on the figure. Both are looking at the top of a 15-metre tree. From point A, the angle of elevation is 65. What is the angle of elevation of the bird watcher at position B? Show all your work. Michel s room is on the second floor of his house. His neighbour lives in a flat-roofed building 15 metres away. From his window, Michel sees the top corner of his neighbour s building at an angle of elevation of 28, as shown in the diagram on the right. The angle of depression at which he sees the bottom corner of his neighbour s building is 40. What is the height of this building? Show all your work. A 65? 4 m B m 40 h 2 h 1 D C 15 m

38 22 13 Julio wants to purchase a waterfront property on Lake Malartic. The surveyor gave him a plan of the dimensions of the property. The asking price is $1.35 per square metre. How much would it cost Julio to buy this property? Show all your work. Triangle ABC shown on the right represents a plot of land in which: m AC = 64.3 m m AB = 50 m m B = 100 m BAC = 30 To the nearest m 2, what is the area of the plot of land? A 100 m 50 m m 50 m 64.3 m B m C

39 14 Krystal, standing at point A, uses a clinometer to determine the angles of elevation of her window and the roof of her house. These are 20 and 40 respectively. She knows that the top of her window is 4 m above the ground. This situation is represented in the diagram below. To the nearest tenth of a metre, how far is it from the top of her window to the roof of her house? A m? Roof Window

40 16 Given triangle ABC with right angle B. What is the measure of segment BD? Show your work. A 15 cm B D 53 C

41 17 18 Determine the area of triangle ABD illustrated in the diagram below. Show all your work. A 50 5 cm C D 40 Without doing any calculations, explain how you would determine that the height of BD relative to AC is cm. A 26 cm 35 cm B D cm B C

42 19 The diagram below represents the roof of a barn. The measure of angle BAC is What is the measure of angle ABC given that BC measures 1.5 m, AB measures 2 m and AC measures 1 m? Show all the work needed to solve the problem. A D C B

43 20 From point A, Anthony observes a bird at the top of the tree at an angle of 20. From point S, Steven observes the same bird at an angle of 40. Steven is 10 metres from the foot of the tree. What is the distance between Anthony and Steven? Show all your work A S P 10 m O

44 21 Mara has been given the job of decorating her company s staff room. She wants to draw a triangle on one of the walls and paint it blue. A sketch of the wall with the measures of the triangle is shown below. Mara must find the area of the blue triangle before buying the paint. Each litre of paint covers 2 square metres. The paint she wants is sold only in 1-litre containers. How many litres of paint will Mara have to buy? Show all your work m 40