Pythagoras' theorem

The diagram shows cuboid ABCDEFGH

AB = 9 cm AF = 7 cm FC = 18 cm

Calculate the length of BC Give your answer correct to 3 significant figures.

The diagram shows two triangles, ADE and CDB

ABD is a straight line.

AE = 28 cm ED = 45 cm AB = 21 cm CD = 35 cm

angle AED = angle CBD = 90°

Work out the area of triangle CDB Give your answer correct to 3 significant figures.

ABD and ABC are right‑angled triangles.

AB = 16 cm BC = 12 cm AD = 1.5AC

Find the length of CD Give your answer correct to 3 significant figures.

ABD and ABC are right‑angled triangles.

AB = 16 cm BC = 12 cm AD = 1.5AC

Find the length of CD Give your answer correct to 3 significant figures.

The diagram shows a square-based pyramid ABCDE

EA = EB = EC = ED

M is the centre of the horizontal square base ABCD Q is the midpoint of AB Angle EQM 80

EA: AB n:1

Find the value of n Give your answer correct to 3 significant figures.

The diagram shows a shape ABCDE made from a right-angled triangle ABE and a

ABC is a straight line.

AB = 15 cm AE = 17 cm

The perimeter of triangle ABE is 40 cm

Work out the area of the shape ABCDE

The diagram shows a trapezium, ABCD

DAB and ADC are right angles.

AD = 15 cm DC = 14 cm

The area of the trapezium is 360 cm²

Work out the perimeter of the trapezium.

The diagram shows a roof support.

The roof support is made from four lengths of wood, AB, AC, BC and MC

AC = BC = 9 m AB = 12 m

angle AMC = 90°

Lewis is going to buy lengths of wood to make the roof support.

The wood costs 21.50 euros per metre. Each length of wood he buys has to be a whole number of metres.

Work out the total cost of the wood Lewis needs to buy. Show your working clearly.

The diagram shows a trapezium, ABCD

DAB and ADC are right angles.

AD = 15 cm DC = 14 cm

The area of the trapezium is 360 cm²

Work out the perimeter of the trapezium.

The diagram shows a roof support.

The roof support is made from four lengths of wood, AB, AC, BC and MC

AC = BC = 9 m AB = 12 m

angle AMC = 90°

Lewis is going to buy lengths of wood to make the roof support.

The wood costs 21.50 euros per metre. Each length of wood he buys has to be a whole number of metres.

Work out the total cost of the wood Lewis needs to buy. Show your working clearly.

The diagram shows a cuboid ABCDEFGH with horizontal base ADEH

M is the midpoint of the base ADEH and P is the midpoint of the edge CF

Work out the size of angle BMP Give your answer correct to one decimal place.

The diagram shows a hexagon ABCDEF

Angle BCF = 30° AB, FC and ED are parallel.

Calculate the area of ABCDEF Show your working clearly.

Here is triangle ABC

Work out the value of x Give your answer correct to one decimal place. Show your working clearly.

The diagram shows the positions of three villages, A, B and C

The bearing of B from A is 054° The bearing of C from B is 132°

Melur walks from A to B She then walks from B to C and from C to A

Melur walks at an average speed of 6 km/h

Work out the total time Melur takes. Give your answer in hours and minutes.

[In this question 1 cm = 1 unit on the x-axis and 1 cm = 1 unit on the y-axis]

P is a point on a circle with centre (0, 0) The coordinates of P are (8, 10)

The line L is the tangent to the circle at the point P L crosses the x-axis at the point Q and crosses the y-axis at the point R

Work out the length of RQ Give your answer correct to 3 significant figures.

The diagram shows equilateral triangle ABC with sides of length 10 cm. A circle is drawn inside the triangle.

D, E and F are points on the circle.

ADB, BEC and CFA are tangents to the circle.

Calculate the total area of the regions shown shaded in the diagram. Give your answer correct to 3 significant figures.

The diagram shows a shaded shape AEBCD made by removing triangle AEB from

AE = 7.2 cm BE = 5.4 cm BC = 6 cm angle AEB = 90°

Work out the perimeter of the shaded shape.

The diagram shows a shape made up of three semicircles, enclosing a right‑angled triangle.

AB, BC and CA are each the diameter of a semicircle.

BC = CA = 6 cm.

Work out the perimeter of the shape. Give your answer correct to one decimal place.

The diagram shows a shape made up of three semicircles, enclosing a right‑angled triangle.

AB, BC and CA are each the diameter of a semicircle.

BC = CA = 6 cm.

Work out the perimeter of the shape. Give your answer correct to one decimal place.

The diagram shows a shaded shape AEBCD made by removing triangle AEB from

AE = 7.2 cm BE = 5.4 cm BC = 6 cm angle AEB = 90°

Work out the perimeter of the shaded shape.

The diagram shows right-angled triangle ABD

AB = 14 cm AD = 8 cm

C is the point on BD such that angle BAC = 38°

Work out the length of CD Give your answer correct to 3 significant figures.

The diagram shows right-angled triangle ABD

AB = 14 cm AD = 8 cm

C is the point on BD such that angle BAC = 38°

Work out the length of CD Give your answer correct to 3 significant figures.

The diagram shows an isosceles triangle ABC

AB = 7 cm AC = BC = ycm

The area of the triangle is 42 cm²

Work out the value of y

R and T are points on a circle, centre O

RT = 12 cm M is the midpoint of RT Angle ROM = 52

Work out the area of the circle. Give your answer correct to 3 significant figures.

The diagram shows a triangle ABC inside a semicircle.

A, B and C are points on the semicircle.

AB is the diameter of the semicircle.

Angle ACB = 90° Angle BAC = 50° AC = 18 cm

Work out the perimeter of the semicircle. Give your answer correct to 2 significant figures.

The diagram shows a triangular prism, ABCDEF, with a rectangular base ABCD

AB = 6 cm DE = 2.2 cm angle DAE = 18 angle ADE = 90

Work out the angle that BE makes with the plane ABCD Give your answer correct to one decimal place.

The diagram shows an isosceles triangle, with base length 24 cm.

The perimeter of the triangle is 54 cm.

Work out the area of the triangle.

The diagram shows a solid prism ABCDEFGHIJ

The prism is such that each cross section is a pentagon where

AE = BC = x cm AB = 2 x cm ED = CD = 8 cm

angle EAB = angle CBA = 90° angle AED = angle BCD = 120°

Given that AG = BH = EF = DJ = CI = 12 cm

calculate the angle that AJ makes with the base ABHG of the prism. Give your answer correct to 3 significant figures.

The diagram shows a solid cone and a solid sphere.

The cone has base radius r, slant height l and perpendicular height h The sphere has radius r

The base radius of the cone is equal to the radius of the sphere.

Given that

k volume of the cone = volume of the sphere

show that the total surface area of the cone can be written in the form

where a is a constant to be found.

25Two circles, C₁and C₂, are drawn on a centimetre grid, with a scale of 1 cm for 1 unit on each axis.

The centre of circle C₁ is at the point with coordinates (–1, 3) and the radius of C₁ is 13 cm.

The centre of circle C₂ is at the point with coordinates (7, 18) and the radius of C₂ is 6 cm.

Work out the distance between the centre of C₁and the centre of C₂

Explain why circle C₁intersects circle C₂

10Two circles, C₁ and C₂, are drawn on a centimetre grid, with a scale of 1 cm for 1 unit on each axis.

The centre of circle C₁ is at the point with coordinates (–1, 3) and the radius of C₁ is 13 cm.

The centre of circle C₂ is at the point with coordinates (7, 18) and the radius of C₂ is 6 cm.

Work out the distance between the centre of C₁and the centre of C₂

Explain why circle C₁intersects circle C₂

The diagram shows a quadrilateral ABCD

In the diagram, ABC and DAC are right‑angled triangles.

BC = 6 cm AC = 7.5 cm

The area of quadrilateral ABCD is 31.5 cm²

Work out the length of AD

The diagram shows a cube ABCDEFGH with sides of length 6 cm.

T is the midpoint of AB and V is the midpoint of CH

Work out the distance from T in a straight line through the cube.

The diagram shows triangle ABP inside the regular hexagon ABCDEF

AB = 5 cm BP = 2 cm Angle ABP = 90

Work out the size of angle PAF Give your answer correct to 3 significant figures.

The diagram shows triangle ABP inside the regular hexagon ABCDEF

AB = 5 cm BP = 2 cm Angle ABP = 90

Work out the size of angle PAF Give your answer correct to 3 significant figures.

The diagram shows isosceles triangle ABC

AB = AC = 17.5 cm
BC = 28 cm

Calculate the area of triangle ABC

The diagram shows isosceles triangle ABC

AB = AC = 17.5 cm
BC = 28 cm

Calculate the area of triangle ABC

The diagram shows the prism ABCDEFGHJK with horizontal base AEFG

^_`ab$ is a cross section of the prism where ABDE is a square BCD is an equilateral triangle

EF = 2AE M is the midpoint of GF so that JM is vertical.

Angle MAJ = y

In the diagram, ABC is a right-angled triangle and DEF is a semicircular arc.

In triangle ABC

BC = 24 cm angle ABC = 90° angle BCA = 30°

The points D and F lie on AC so that DF is the diameter of the semicircular arc DEF The radius of the semicircular arc is 3 cm.

Work out the length of AFEDC Give your answer correct to 2 significant figures.

In the diagram, ABC is a right-angled triangle and DEF is a semicircular arc.

In triangle ABC

BC = 24 cm angle ABC = 90° angle BCA = 30°

The points D and F lie on AC so that DF is the diameter of the semicircular arc DEF The radius of the semicircular arc is 3 cm.

Work out the length of AFEDC Give your answer correct to 2 significant figures.

A zip wire is shown as the dashed line AC in the diagram.

The zip wire is supported by two vertical posts AB and CD standing on horizontal ground.

CD = 2.6 m BD = 12 m

The zip wire makes an angle x with the horizontal, as shown in the diagram. The design of the zip wire requires the angle x to be at least 5°

Work out the least possible height of the post AB Give your answer correct to 3 significant figures.

ABCD is a horizontal rectangular field.

A vertical pole, AE, is placed at the corner A of the field.

AE = 12 m AB = 18 m AD = 8 m

Calculate the size of the angle between EC and the plane ABCD Give your answer correct to one decimal place.

Here is a right-angled triangle.

Calculate the value of h. Give your answer correct to 3 significant figures.

Here is a right-angled triangle.

Calculate the value of h. Give your answer correct to 3 significant figures.

Here is isosceles triangle ABC.

D is the midpoint of AC and DB = 16 cm.

Angle DAB = 65

Work out the perimeter of triangle ABC. Give your answer correct to one decimal place.

Here is isosceles triangle ABC.

D is the midpoint of AC and DB = 16 cm.

Angle DAB = 65

Work out the perimeter of triangle ABC. Give your answer correct to one decimal place.

The diagram shows the positions of three ships, A, B and C.

Ship B is due north of ship A.

The bearing of ship C from ship A is 120

Calculate the bearing of ship C from ship B. Give your answer correct to the nearest degree.

A, B and C are points on a circle with centre O.

AOC is a diameter of the circle.

AB = 8 cm BC = 15 cm

Angle ABC = 90

Work out the total area of the regions shown shaded in the diagram. Give your answer correct to 3 significant figures.

ABC is an isosceles triangle in a horizontal plane. The point T is vertically above B.

Angle ABC = 140 AB = BC = 8 cm TB = 10 cm M is the midpoint of AC.

Calculate the size of the angle between MT and the horizontal plane ABC. Give your answer correct to one decimal place.

The diagram shows trapezium ABCD in which BC and AD are parallel.

The trapezium has exactly one line of symmetry.

BC = 8.4 cm AD = 17.6 cm

The trapezium has area 179.4 cm²

Work out the size of angle ABC. Give your answer correct to 1 decimal place.

The diagram shows a prism ABCDEFGH with a horizontal base.

The base of the prism, EFGH, is a square of side 12 cm. Trapezium ADEF is a cross section of the prism where AF and DE are vertical edges.

DE = CH = 10 cm AD = BC = 15 cm

Work out the size of the angle between CF and the base EFGH. Give your answer correct to one decimal place.

Work out the length of BE. Give your answer correct to one decimal place.

The diagram shows a regular octagon ABCDEFGH.

Each side of the octagon has length 10 cm.

Find the area of the shaded region ACDEH. Give your answer correct to the nearest cm²

Markus makes a steel framework. The framework is in the shape of the right‑angled triangle ABC shown in the diagram.

The steel that Markus uses costs $22 per metre. The steel can only be bought in a length that is a whole number of metres.

Work out the total cost of the steel that Markus buys in order to make the framework.

The diagram shows a right-angled triangle.

Calculate the value of x. Give your answer correct to one decimal place.

The diagram shows four congruent right‑angled triangles ABJ, BCI, CDH and DEG. The diagram also shows the straight line ABCDEF. Diagram NOT

AJ = 15 cm Angle BAJ = 35

AF = 80 cm

Work out the length of EF. Give your answer correct to 3 significant figures.

The diagram shows cuboid ABCDEFGH.

AB = 5 cm AH = 4 cm The size of the angle between CH and the plane ABCD is 35°

Calculate the volume of the cuboid. Give your answer correct to 3 significant figures.

Here is a right-angled triangle.

Calculate the length of PQ. Give your answer correct to 3 significant figures.

The diagram shows two right-angled triangles, DEF and EFG.

Work out the length of EG. Give your answer correct to 3 significant figures.

The diagram shows a cuboid ABCDEFGH.

EH= 9 cm, HG= 5 cm and GB= 6 cm.

Work out the size of the angle between AH and the plane EFGH. Give your answer correct to 3 significant figures.

H is the point with coordinates (−4, 1) J is the point with coordinates (j, 15) where j <0 K is the point with coordinates (6, k)

M is the midpoint of JK. The gradient of HM is 2

Find the value of j and the value of k.

The diagram shows a rectangle and a diagonal of the rectangle.

Work out the length of the diagonal of the rectangle. Give your answer correct to 1 decimal place.

The diagram shows a shaded shape ABCD made from a semicircle ABC and a right‑angled triangle ACD.

AC is the diameter of the semicircle ABC.

Work out the perimeter of the shaded shape. Give your answer correct to 3 significant figures.

The diagram shows a rectangle and a diagonal of the rectangle.

Work out the length of the diagonal of the rectangle. Give your answer correct to 1 decimal place.

The diagram shows a shape made from a right-angled triangle and a semicircle.

AC is the diameter of the semicircle. BA = BC = 6 cm Angle ABC = 90°

Work out the area of the shape. Give your answer correct to 1 decimal place.

The diagram shows isosceles triangle ABC.

AB = AC = 7.5 cm. The height of the triangle is 6 cm.

Calculate the area of the triangle.

ABCD is a trapezium.

21.2 cm C

Calculate the perimeter of the trapezium. Give your answer correct to 3 significant figures.

The diagram shows cuboid ABCDEFGH.

For this cuboid

the length of AB: the length of BC: the length of CF = 4: 2: 3

Calculate the size of the angle between AF and the plane ABCD. Give your answer correct to one decimal place.

The diagram shows a solid pyramid ABCDE with a horizontal base.

The base, BCDE, of the pyramid is a square of side 10 cm.

The vertex A of the pyramid is vertically above the centre O of the base so that AB = AC = AD = AE

The total surface area of the pyramid is 360 cm²

Work out the size of the angle between AC and the base BCDE. Give your answer correct to 3 significant figures.

M, N and P are points on a circle, centre O. MON is a diameter of the circle.

MP = 3.5 cm PN = 9.7 cm

Angle MPN = 90°

Work out the circumference of the circle. Give your answer correct to 3 significant figures.

M, N and P are points on a circle, centre O. MON is a diameter of the circle.

MP = 3.5 cm PN = 9.7 cm

Angle MPN = 90

Work out the circumference of the circle. Give your answer correct to 3 significant figures.

Work out the value of x. Give your answer correct to 3 significant figures.

From point A, Stanley walks 200 m due east to point B. From B, he then walks 160 m due south to point C.

Work out the length of AC. Give your answer correct to 3 significant figures.

ABCD is a trapezium.

Work out the size of angle x. Give your answer correct to 1 decimal place.

From point A, Stanley walks 200 m due east to point B. From B, he then walks 160 m due south to point C.

Work out the length of AC. Give your answer correct to 3 significant figures.

The diagram shows a triangular prism.

AF = 10 cm, AB = 24 cm and BC = 8 cm. Angle FAB = angle ADC = angle BCD = 90

Work out the size of the angle between the line BE and the plane ABCD. Give your answer correct to 1 decimal place.