PLC Papers Created For:

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1 PLC Papers Created For: Year 10 Topic Practice Papers: Pythagoras and Trigonometry

2 Pythagoras 1 Grade 5 Objective: Know and use Pythagoras's theorem for right-angled triangles Question 1 ABC is a right angled triangle. AB = 9 cm, BC = 12 cm Calculate the length of AC. Question 2 (Total 3 marks). (3) ABC is a right angled triangle. AB = 11 cm, AC = 18 cm Calculate the length of BC. Give your answer correct to 1 decimal place.. (Total 3 marks) (3)

3 Question 3 ABCD is a rectangle. AB = 19 m, AD = 13 m Work out the length of the diagonal BD. Give your answer correct to 3 significant figures. (Total 4 marks). (4) Total /10

4 Pythagoras 2 Grade 5 Objective: Know and use Pythagoras's theorem for right-angled triangles Question 1 ABC is a right angled triangle. AB = 8 m, BC = 14 m Calculate the length of AC. Give your answer correct to 1 decimal place. Question 2 (Total 3 marks). (3) ABC is a right angled triangle. AB = 10 cm, AC = 21 cm Calculate the length of BC. Give your answer correct to 1 decimal place.. (Total 3 marks) (3)

5 Question 3 ABCD is a rectangle. AB = 23 m, AD = 12 m Work out the length of the diagonal BD. Give your answer correct to 3 significant figures. (Total 4 marks). (4) Total marks / 10

6 Pythagoras 3 Grade 5 Objective: Know and use Pythagoras's theorem for right-angled triangles Question 1 This is a funicular railway in Portugal. The track length is 274m and the station at the top of the hill is 116m higher than the station at the bottom. How far does the train travel horizontally to the nearest metre?. (Total 3 marks) Question 2 Jack runs around the perimeter of a rectangular field that measures 120m by 46m. On Sunday he decides to run the length, followed by the width, but then cut back across the diagonal. How much shorter is this route than the whole perimeter? Give your answer correct to the nearest metre. (3). (Total 4 marks) (4)

7 Question 3 Ben s tent has its end in the shape of an isosceles triangle. The vertical height is 1.9 m and the horizontal base measurement is 2.1 m. What is the length of the sloping edge of the front end? Give your answer correct to 3 significant figures.... (3) Total marks / 10

8 Pythagoras 4 Grade 5 Objective: Know and use Pythagoras's theorem for right-angled triangles Question 1 Amy stands a 5m ladder against a vertical wall. The foot of the ladder is 1.8 m from the bottom of the wall. How far up the wall will the ladder reach? Give your answer to one decimal place. (Total 3 marks) Question 2 A rectangular gate in a field is 2.5 m wide and 1.1 m high. It is made up of five horizontal pieces of wood each 2.5m long, two vertical pieces (one at each end) each 1.1 m long and a diagonal piece to give the gate strength. What is the total length of wood that must be bought to build this gate, if the wood can only be bought in whole metre lengths... (3). (Total 4 marks) (4)

9 Question 3 The diagram shows how a ramp for wheelchairs is to be placed over three steps. What will be the length of the ramp to two significant figures? (Total 3 marks)... (3) Total marks / 10

10 Pythagoras and Trigonometry 2D and 3D 2 Grade 7 Objective: Solve problems using Pythagoras's theorem and trigonometry in general 2-D triangles and 3-D figures Question 1. The diagram represents a cuboid ABCDEFGH. Its height is 2.5metres and its width is 4 metres. Angle GHF = 62 o Diagram NOT drawn accurately (a) Calculate the length of the diagonal HF. Give your answer to one decimal place. (b) Calculate the angle CHF. Give your answer to one decimal place (2) (2) Question 2. ABC is an isosceles triangle. AC = 18cm Vertical height = 14cm Calculate angle BCA to 1dp. B 14cm (Total 4 marks) Diagram NOT drawn accurately A 18cm C (2 marks)

11 Question 3. ABCDE is a square based pyramid. The base has sides 9cm. Diagram NOT drawn accurately The vertical height of the pyramid is 8cm. (a) Calculate the length of AC. Give your answer correct to one decimal place. (1) (b) Calculate the length of AE. Give your answer correct to one decimal place. (1) (c) Calculate the size of angle EAC. (2) Total /10

12 Pythagoras and Trigonometry 2D and 3D 4 Grade 7 Objective: Solve problems using Pythagoras's theorem and trigonometry in general 2-D triangles and 3-D figures Question 1. ACDEF is a tent. Diagram NOT drawn accurately The base is 2.2m wide and 3.6m long. The ends are isosceles triangles. The ends are at an angle of 80 o to the base. Angle AEB and angle DFC is 70 o. M is the midpoint of AB. What is the maximum height inside the tent?... (Total 3 marks) Question 2. Ed wants to fence his new triangular shaped paddock ABC He knows the widest part is 5m. B Diagram NOT drawn accurately He knows the longest part is 8m. He knows the two diagonal sides are the same length. 8m Fencing costs 3.99 per metre. A 5m C What will be the cost of fencing the paddock?... (Total 2 marks)

13 Question 3. Ben is 1.62m tall. The tent he is considering buying is a square based pyramid. The length of the base is 3.2m. 2m The poles AE, CE, AE and BE are 2m long. 2.2m Ben wants to know if he will be able to stand up in the middle of the tent. Explain your answer clearly. (3) What will be the angle between the poles and the base of the tent? (2) Total /10

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15 Pythagoras and Trigonometry 2D and 3D 1 Grade 7 Objective: Solve problems using Pythagoras's theorem and trigonometry in general 2-D triangles and 3-D figures Question 1. ABC is an isosceles triangle BC = 24cm Vertical height = 20cm B 20cm 24cm Diagram NOT drawn accurately A Calculate the length of AC. Give your answer correct to one decimal place. C (Total 2 marks) Question 2. ABCDEFGH is a cuboid AE = 5cm AB = 6cm Diagram NOT drawn accurately BC = 9cm (a) Calculate the length of AG. Give your answer correct to 3 significant figures. (b) Calculate the size of the angle between AG and the face ABCD. Give your answer correct to 1 decimal place. (1) (3) (Total 4 marks)

16 Question 3. The diagram shows a square based pyramid. The square base has sides 18cm. Diagram NOT drawn accurately 18cm (a) Calculate the length of the diagonal AB. Give your answer correct to 1 decimal place. (1) (b) If VBA = 58 o, calculate the vertical height VC. Give your answer correct to 1 decimal place. (3) (Total 4 marks) Total /10

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18 Pythagoras and Trigonometry 2D and 3D 3 Grade 7 Objective: Solve problems using Pythagoras's theorem and trigonometry in general 2-D triangles and 3-D figures Question 1. A piece of land is the shape of an isosceles triangle with sides 7.5m, 7.5m and 11m. Turf can be bought for per 5m 2 roll. How much will it cost to turf the piece of land? (Total 3 marks) Question 2. ABCDEFGH is a cuboid shaped cardboard box. Length = 18cm Width = 12cm Height = 6cm 6cm 12cm 18cm (a) Calculate AC the diagonal length of the base of the box. (b) Harry the Magician has promised to post his spare magic wand to a friend. His spare magic wand is 22cm long. Explain whether or not he could use this box to post the wand. (1) (2) (Total 3 marks)

19 Question 3. ABCDEF is a wedge shaped skate ramp. AB = 3m BC = 4m FC = 2m 3m 4m 2m (a) If Owen wants to skate from corner E at the top of the ramp to corner B at the bottom, what is the shortest distance he can travel? (2) (b) The angle of elevation of the ramp enables a judge to categorise its difficulty. Category A ramps have an angle of elevation less than 20 o. Category B ramps have an angle of elevation between 20 o and 30 o inclusive. Category C ramps have an angle of elevation greater than 30 o. Explain what type of ramp ABCDEF is. (2) (Total 4 marks) Total /10

20 PLC Papers Created For: Year 10 Topic Practice Papers: Pythagoras and Trigonometry

21 Pythagoras 1 Grade 5 Solutions Objective: Know and use Pythagoras's theorem for right-angled triangles Question 1 ABC is a right angled triangle. AB = 9 cm, BC = 12 cm Calculate the length of AC. AC 2 = (M2 square, add) = = 225 AC = 15 (A1) Question 2 (Total 3 marks) 15cm. (3) ABC is a right angled triangle. AB = 11 cm, AC = 18 cm Calculate the length of BC. Give your answer correct to 1 decimal place. BC 2 = (M2 square, subtract) = = 203 BC = 14.2 (A1) 14.2 cm. (Total 3 marks) (3)

22 Question 3 ABCD is a rectangle. AB = 19 m, AD = 13 m Work out the length of the diagonal BD. Give your answer correct to 3 significant figures. BD 2 = (M2 square, add) = = 530 BD = 23.0 (A2 correct, correct to 3sf) (Total 4 marks) Total / m. (4) Total marks / 10

23 Pythagoras 2 Grade 5 Solutions Objective: Know and use Pythagoras's theorem for right-angled triangles Question 1 ABC is a right angled triangle. AB = 8 m, BC = 14 m Calculate the length of AC. Give your answer correct to 1 decimal place. AC 2 = (M2 square, add) = = 260 AC = 16.1 (A1) Question 2 (Total 3 marks) 16.1 m. (3) ABC is a right angled triangle. AB = 10 cm, AC = 21 cm Calculate the length of BC. Give your answer correct to 1 decimal place. BC 2 = (M2 square, subtract) = = 341 BC = 18.5 (A1) 18.5 cm. (Total 3 marks) (3)

24 Question 3 ABCD is a rectangle. AB = 23 m, AD = 12 m Work out the length of the diagonal BD. Give your answer correct to 3 significant figures. BD 2 = (M2 square, add) = = 673 BD = 25.9 (A2 correct, correct 3sf) (Total 4 marks) Total / m. (4)

25 Pythagoras 3 Grade 5 Solutions Objective: Know and use Pythagoras's theorem for right-angled triangles Question 1 This is a funicular railway in Portugal. The track length is 274m and the station at the top of the hill is 116m higher than the station at the bottom. How far does the train travel horizontally to the nearest metre? horizontal 2 = (M2 subtract and squared) = = horizontal = (A1) 248 m. (Total 3 marks) Question 2 Jack runs around the perimeter of a rectangular field that measures 120m by 46m. On Sunday he decides to run the length, followed by the width, but then cut back across the diagonal. How much shorter is this route than the whole perimeter? Give your answer correct to the nearest metre. (3) diagonal 2 = (M1) = = diagonal = (A1) Perimeter = 2(120+46) = 332 Short run = =313 Difference = = 19 (M1 A1) 19m. (Total 4 marks) (4)

26 Question 3 Ben s tent has its end in the shape of an isosceles triangle. The vertical height is 1.9 m and the horizontal base measurement is 2.1 m. What is the length of the sloping edge of the front end? Give your answer correct to 3 significant figures. Half base = 1.05 (M1) height 2 = (M1) = = height = 2.17 (A1) (Total 3 marks) 2.17 m. (3) Total marks / 10

27 Pythagoras 4 Grade 5 Solutions Objective: Know and use Pythagoras's theorem for right-angled triangles Question 1 Amy stands a 5m ladder against a vertical wall. The foot of the ladder is 1.8 m from the bottom of the wall. How far up the wall will the ladder reach? Give your answer to one decimal place. vertical 2 = = = vertical = 4.66 (M2 subtract, square) (Total 3 marks) Question 2 A rectangular gate in a field is 2.5 m wide and 1.1 m high. It is made up of five horizontal pieces of wood each 2.5m long, two vertical pieces (one at each end) each 1.1 m long and a diagonal piece to give the gate strength. What is the total length of wood that must be bought to build this gate, if the wood can only be bought in whole metre lengths. (A1) 4.7 m. (3) diagonal 2 = = = 7.46 diagonal = 2.73 (A1) (M1) Horizontal lengths = 5 x 2.5 = 12.5 Vertical = 2 x 1.1 = 2.2 Diagonal = 2.73 Total = (M1) (A1) 18 m. (Total 4 marks) (4)

28 Question 3 The diagram shows how a ramp for wheelchairs is to be placed over three steps. What will be the length of the ramp to two significant figures? Length of each step 2 = (M1) = = 2260 Length = So three steps = 3x (M1) = 140 (A1) (Total 3 marks) 140 cm. (3) Total marks / 10

29 Pythagoras and Trigonometry 2D and 3D 2 Grade 7 Solutions Objective: Solve problems using Pythagoras's theorem and trigonometry in general 2-D triangles and 3-D figures Question 1. The diagram represents a cuboid ABCDEFGH. Its height is 2.5metres and its width is 4 metres. Angle GHF = 62 o Diagram NOT drawn accurately (a) Calculate the length of the diagonal HF. Give your answer to one decimal place. (b) Calculate the angle CHF. Give your answer to one decimal place cos62 = 4 HF (M1 using cos62) HF = 4 cos62 = 8.5m (A1) (2) tanchf = (M1 Using tanθ) CHF = tan -1 ( ) = 16.4m (A1) FT from (a) (2) Question 2. ABC is an isosceles triangle AC = 18cm Vertical height = 14cm B 14cm (Total 4 marks) Diagram NOT drawn accurately Calculate the angle BCA to 1dp. A 18cm C TanBCA = 14 9 (M1 use of Tan) Tan -1 (14 9) = 57.3 o... (1 mark)

30 Question 3. ABCDE is a square based pyramid. Diagram NOT drawn accurately The base has sides 9cm. The vertical height of the pyramid is 8cm. (a) Calculate the length of AC. Give your answer correct to one decimal place. AC = ( ) = 12.7cm (B1) (1) (b) Calculate the length of AE. Give your answer correct to one decimal place. AE = ( ) = 10.2 cm (B1) (2) (c) Calculate the size of angle EAC. CosEAC = AC AE (M1 use of Cos) EAC = Cos -1 (AC AE) = 51.5 o (A1) (2) Total /10

31 Pythagoras and Trigonometry 2D and 3D 4 Grade 7 Solutions Objective: Solve problems using Pythagoras's theorem and trigonometry in general 2-D triangles and 3-D figures Question 1. ACDEF is a tent. Diagram NOT drawn accurately The base is 2.2m wide and 3.6m long. The ends are isosceles triangles. The ends are at an angle of 80 o to the base. Angle AEB and angle DFC is 70 o. M is the midpoint of AB. E What is the maximum height inside the tent? 35 o tan35 = 1.1 EM E A 1.1m M EM = 1.57m (M1) 1.57 m sin80 = EX 1.57 (M1) Maximum height = 1.55m (A1) 80 o M X... (Total 3 marks) Question 2. Ed wants to fence his new triangular shaped paddock ABC He knows the widest part is 5m. B Diagram NOT drawn accurately He knows the longest part is 8m. He knows the two diagonal sides are the same length. 8m Fencing costs 3.99 per metre. A 5m C What will be the cost of fencing the paddock? AB = = 8.38m (M1) Perimeter of paddock = 21.76m so cost = 22 x 3.99 = (A1)... (Total 2 marks)

32 Question 3. Ben is 1.62m tall. The tent he is considering buying is a square based pyramid. The length of the base is 3.2m. 2m The poles AE, CE, AE and BE are 2m long. 2.2m Ben wants to know if he will be able to stand up in the middle of the tent. Explain your answer clearly. DB = ( ) = 3.1m (M1) Height = ( ) = m (M1) Ben will be able to stand up in the tent (A1) (3) What will be the angle between the poles and the base of the tent? tanθ = (M1) θ = 45.9 o (A1) E D θ 1.55m 1.6m X (2) Total /10

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34 Pythagoras and Trigonometry 2D and 3D 1 Grade 7 Solutions Objective: Solve problems using Pythagoras's theorem and trigonometry in general 2-D triangles and 3-D figures Question 1. ABC is an isosceles triangle BC = 24cm Vertical height = 20cm B 20cm 24cm Diagram NOT drawn accurately A Calculate the length of AC. Give your answer correct to one decimal place. C (= 176) (M1) = 26.5cm (A1) (Total 2 marks) Question 2. ABCDEFGH is a cuboid AE = 5cm AB = 6cm Diagram NOT drawn accurately BC = 9cm (a) Calculate the length of AG. Give your answer correct to 3 significant figures. AG = ( ) = ( ) = 142 = 11.9cm (1) (b) Calculate the size of the angle between AG and the face ABCD. Give your answer correct to 1 decimal place. Use of Sin (M1) Sinθ = = (M1 ft from (a) ) Θ = 24.8 o (A1) A θ 142 G 5 C (3) (Total 4 marks)

35 Question 3. The diagram shows a square based pyramid The square base has sides 18cm Diagram NOT drawn accurately 18cm (a) Calculate the length of the diagonal AB. Give your answer correct to 1 decimal place. AB = ( ) AB = 18 2 cm = 25.5 cm (1) (b) If VBA = 58 o, calculate the vertical height VC. Give your answer correct to 1 decimal place. Use of tan58 o (M1) tan58 o = VC 9 2 (M1 ft from their 18 2) VC = 9 2 tan58 VC = 20.4cm 1M (A1) (3) (Total 4 marks) Total /10

36 Pythagoras and Trigonometry 2D and 3D 3 Grade 7 Solutions Objective: Solve problems using Pythagoras's theorem and trigonometry in general 2-D triangles and 3-D figures Question 1. A piece of land is the shape of an isosceles triangle with sides 7.5m, 7.5m and 11m. Turf can be bought for per 5m 2 roll. How much will it cost to turf the piece of land? ( ) = 5.10m length of land (M1) Area of land = 11 x = 28.05m 2 (M1) Need to buy 6 rolls at 5.99 per roll Total cost = (A1) (Total 3 marks) Question 2. ABCDEFGH is a cuboid shaped cardboard box. Length = 18cm Width = 12cm Height = 6cm 6cm 12cm 18cm (a) Calculate AC the diagonal length of the base of the box. ( ) = 21.6cm (B1) (b) Harry the Magician has promised to post his spare magic wand to a friend. His spare magic wand is 22cm long. Explain whether or not he could use this box to post the wand. (1) AG = ( ) = 22.4cm (M1) Yes Harry can use the box to post his wand (A1) (2) (Total 3 marks)

37 Question 3. ABCDEF is a wedge shaped skate ramp. AB = 3m BC = 4m FC = 2m 3m 4m 2m (a) If Owen wants to skate from corner E at the top of the ramp to corner B at the bottom, what is the shortest distance he can travel? BD = 5m (M1) Pythagorean triplet or use of Pythagoras theorem EB = = 5.4m (A1) (2) (b) The angle of elevation of the ramp enables a judge to categorise its difficulty. Category A ramps have an angle of elevation less than 20 o. Category B ramps have an angle of elevation between 20 o and 30 o inclusive. Category C ramps have an angle of elevation greater than 30 o. Explain what type of ramp ABCDEF is. Tanθ = 2 4 (M1) Use of Tan θ = 26.6 o so a category B ramp (A1) (2) (Total 4 marks) Total /10