Source: Questions from Complete International Mathematics For Cambridge IGCSE - David Rayner, Jim Fenson

1. A ladder of length 8 m rests against a wall so that the angle 
    between the ladder and the wall is 31°. 
    How far is the base of the ladder from the wall? 
Let the base of ladder be x m from the wall
The ladder, the wall and the floor form a triangle

   The base of the ladder is 4.1 m from the wall.

2. A ship sails 200 km on a bearing of 243.7°.
    a) How far south has it traveled?
    b) How  far west has it traveled? 

Let OA be the line traveled by the ship.
N, E, S and W represent the North,
East, south and West directions respectively
Bearing of ship = 243.7°( The line of travel of
ship makes an angle of 243.7° measured
clockwise from North)
North-South is a straight line = 180°

   The Ship has traveled 88.6 km to the south and 179.3 km to the west.

3. An aircraft flies 500 km on a bearing of 100° and then 600 km on a bearing of 160°. 
    Find the distance and bearing of the finishing point from the starting point.
Let the starting point be O.
The aircarft flies 500 km on bearing of 100°.
It reaches point A
Then 600 km on a bearing of 160°. 
Let the finishing point be X
Bearing is 100° i.e the total angle measured
clockwise from North is 100°.

The horizontal at O represents West.
Therefore the angle between the line
of travel of ship and West = 100°-90°=10°

The bearing of aircraft is now 160°.
Hence the angle it makes with the
horizontal at point A is 160°-90°=70°
The total distance traveled along
horizontal axis is a+b
The total distance traveled along
vertical axis is c+d

The Aircraft is at a distance of 953.8 km from the starting point. 
The bearing of the finishing point is 180 - 47 = 133°

4. For the points P(2,5), Q(5,1) and R(0,-3), Calculate the angle PQR


5. From the top of a tower 75 m, a man sees two goats, both due west of him. 
   If the angle of depression of the two goats are 10° and 17° respectively, 

   Calculate the distance between them.

 Angle of depression of goat G1 = 17°
Angle of depression of goat G2 = 10°

The distance between the goats is 180 m

6. A Chord of length 12 cm subtends an angle of 78.6°at the center of the circle. 
    Find the radius of the circle.
Radius of circle = 9.5 cm

7. A rocket flies 10 km vertically, then 20 km at an angle of 15° to the vertical and finally 60 km at an angle of 26° to the vertical. Calculate the vertical height of the rocket at the end of the third stage.
Vertical height of the rocket at the end of third stage =  83.2 km

8. Find x:

9. Diagram A  shows a kitchen floor tiled with regular octagons and squares. 
    Diagram B shows and enlarged view of one octagon.
   The smaller green squares have sides of length 12 cm. 
   Find the width of the octagons.

Interior Angle of a regular Octagon = 135°
Hence angle in triangle = 45°
Width of Octagon = (8.5 X 2) + 12
                             = 29 cm

10. A farm gate is 1.3 m high. 

     The diagonal crossbar has a width of 15 cm and makes an angle of 24° with the horizontal. 

     Find the length of x and y in the diagram. 

     Hence find the width of the gate.

y = 113.6 cm
Width of gate = 255 cm

11. A rectangular paving stone 3 m by 1 m rests against a vertical wall as shown. 

What is the height of the highest point of the stone above ground?

  Point B:Point A:
Highest point of stone above ground = 2.7 + 0.45 = 3.13 m

12. The diagram shows the cross section of a rectangular fish tank. 

    When AB is inclined at 40°, the water just comes up to A. 
    The tank is then lowered so that BC is horizontal. 

    What is now the depth of the water in the tank?

 Figure 1:Figure 2:

The area in cross section of  Figure 1 and
area in cross section of figure 2 will be equal
as the volume of water is the same.
The depth of water when BC is horizontal = 27.12 cm