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

Let the base of ladder be x m from the wall The ladder, the wall and the floor form a triangle

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. 

   Calculate the distance between them.

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

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.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. 

     Hence find the width of the gate.

POINT A

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. 

    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