Geometry and trigonometry: exam review Exam review

A roof truss is a right-angled triangle ABC, with the right angle at B. The base AB is 8.4 m and the height BC is 2.9 m. [7]

1. (a) Find the size of angle BAC. [2]
2. (b) A second roof truss has the same base AB and angle BAC = 25°. Find its height BC. [2]
3. (c) Two straight roads on a map have equations \(y = 3x\) and \(y = -0.5x\). Find the acute angle between the roads. [3]

A triangular plot of land ABC has AB = 7.4 m, AC = 9.1 m and angle BAC = 64°. [7]

1. (a) Find the length of BC. [3]
2. (b) Find the size of angle ABC. [2]
3. (c) Find the area of triangle ABC. [2]

A storage box ABCDEFGH is a cuboid with AB = 80 cm, BC = 50 cm and CG = 40 cm. [5]

1. (a) Find the length of AG. [2]
2. (b) Find the angle between AG and the base ABCD. [3]

A sector OAB of a circle has centre O, radius 9.5 cm and angle AOB = 72°. [5]

1. (a) Find the length of the arc AB. [2]
2. (b) Find the area of the segment between the chord AB and the arc AB. [3]

Mia stands at P and measures the angle of elevation of the top T of a tower as 25°. She walks 45 m directly towards the tower to Q and measures the angle of elevation as 41°. Her eyes are 1.6 m above the ground. N is the point at the height of her eyes directly below T. [5]

1. (a) Find the size of angle PTQ. [1]
2. (b) Use the sine rule in triangle PQT to find QT. [2]
3. (c) Find the height of the tower above the ground. [2]

A triangular garden ABC has AB = 9 m, AC = 12 m and area 40 m². Angle BAC is acute. The point D lies on BC with BD = 5 m. [7]

1. (a) Find the size of angle BAC. [2]
2. (b) Find the length of BC. [2]
3. (c) Find the length of AD. [3]

A goat is tied to the corner P of a rectangular shed, 12 m by 8 m, standing on flat grass. The diagram shows the shed from above. The goat cannot go into the shed. [12]

1. (a) The goat is tied at P by a rope of length 7 m. Find the area of grass the goat can reach. [3]
2. (b) The rope is now 18 m long. After the rope has passed the two corners of the shed, find the radius of each of the two quarter circles the goat can reach beyond them. [2]
3. (c) Find the total area of grass the goat can reach with the 18 m rope. [3]
4. (d) The rope is moved to M, the midpoint of the 12 m wall, and shortened to 10 m. Find the area of grass the goat can now reach. [4]

A rectangular field ABCD has AB = 25 m and AD = 15 m. A goat is tied at A by a rope of length 20 m. The field has a fence on all four sides, and the rope is long enough to reach the side DC. [12]

1. (a) The rope just reaches the side DC at E, and AE = 20 m. Find the length DE and the size of angle EAB. [3]
2. (b) The goat can reach the region ADE and the sector EAF, where F is on AB and AF = AE. Find the total area of the region the goat can reach. [4]
3. (c) Find the percentage of the field that the goat can reach. [2]
4. (d) The rope length r m is changed, where 15 < r < 25. The goat can then reach half of the field. Write down an equation for r and solve it using technology. [3]

A boat leaves port P and sails 14 km on a bearing of 052° to Q. It then sails 9 km on a bearing of 140° to R. [12]

1. (a) Find the size of angle PQR. [2]
2. (b) Find the distance PR. [3]
3. (c) Find the size of angle QPR. [3]
4. (d) Find the bearing of P from R. Give your answer to the nearest degree. [3]
5. (e) Write down the bearing of Q from R. [1]

A glass pyramid VABCD stands in a museum courtyard. The base ABCD is a horizontal square of side 12 m and each sloping edge is 14 m. M is the centre of the base, and V is directly above M. [12]

1. (a) Find the length of AC. [2]
2. (b) Find the height VM of the pyramid. [2]
3. (c) Find the angle between the edge VA and the base ABCD. [2]
4. (d) N is the midpoint of AB. Find the angle between the face VAB and the base. [2]
5. (e) Find the volume of the pyramid. [2]
6. (f) Find the total area of the four triangular faces. [2]

A glass is a cone with its point at the bottom. The radius of the top is 5 cm and the perpendicular height is 12 cm. The glass is held with its axis vertical and filled with juice to a depth of 7.2 cm. [11]

1. (a) Find the volume of the glass when it is full, in cm³ and in litres. [2]
2. (b) Find the radius of the surface of the liquid. [2]
3. (c) Find the volume of the liquid, and write it as a percentage of the volume of the full glass. [3]
4. (d) The liquid is poured into a cylinder with radius 4 cm. Find the depth of the liquid in the cylinder. [2]
5. (e) Instead, the liquid is frozen into a solid hemisphere. Find the radius of the hemisphere. [2]