**UNIVERSITY EXAMINATIONS: 2016/2017**

**EXAMINATION FOR THE CERTIFICATE IN BRIDGING **

**MATHEMATICS**

**MAT102 GEOMETRY**

**DATE: AUGUST, 2017 TIME: 2 HOURS**

**INSTRUCTIONS: Answer Question One & ANY OTHER TWO questions.**

**QUESTION ONE [30 MARKS] – COMPULSORY**

a) To find the height of a tower a surveyor stands some distance away from the base of the tower and

observes the angle of elevation to the top of the tower to be 45°. He then moves 80m nearer and observes

that the angle of elevation to be 60°. Find the height of the tower.

(5 Marks)

b) i) Derive the sine rule (6 Marks)

ii) Solve the triangle ABC using the sine rule given a= 23 cm, c= 18.2 cm and A= 49.32° (7 Marks)

Find the diameter of a circumscribed circle of the triangle ABC given:

C= 100°, B = 50° and c= 90mm (2 Marks)

d) Find the distance measured along the parallel of latitude 56°N, between the two points

(56°N, 23°E ) and (56°N, 17°W). (Give your answer in KMs and nm)

(6 Marks)

e) The interior angle of a hexagon are 2x°, 2/3x°, x +40°, 110°, 130° and 160° .Find the value of the

smallest angle.

(4 Marks)

**QUESTION TWO [20 MARKS]**

a) Calculate the distance in nm and kms between two places along the circle of latitude A (30°N, 20°E)

and B (30°N, 80°E) (5 Marks)

b) A ship in distress sends out a signal and gives its position as (50 N, 20 W). The signal is picked up by

a ship X at (50 N, !5 W) and a ship Y at (55 N, 20 W). Both ships can move at a speed of 20 knots.

i) Find which ship can arrive at the scene first (7 Marks)

ii) Find the speed to the nearest knot, that the other ship should have had to arrive at the scene at the same

time with the earlier ship. (4 Marks)

c) Each interior angle of a regular polygon is 120°. How many sides does it have?

(4 Marks)

**QUESTION THREE [20 MARKS]**

Two towns A and B lie on a latitude 30°N .Their longitudes are 25°W and 95°E respectively. Calculate;

a) The distance from A to B along a parallel of latitude (5 Marks)

b) The shortest distance from A to B along a great circle (5 Marks)

c) A jet takes off from A on Thursday at 1600Hrs and heads for town B at a speed of 800km/hr. It is

refueled and serviced at B for 40 minutes before flying due south to a town C (10°N, 100°E ). At what

time does it land at C? (10 Marks)

**QUESTION FOUR [20 MARKS]**

a) The angles of a quadrilateral are x, 5x, 4x and 2x. Find the value of these angles (6 Marks)

b) Explain THREE characteristics that can prove that two triangles are congruent (6 Marks)

c) Generate the trigonometric ratios of special angles 30° and 60° without calculator computation (hint:

use a triangle)

(8 Marks)

**QUESTION FIVE [20 MARKS]**

A right pyramid has a square base ABCD of sides 8 cms. O is the centre of the base and VO is 10cms,

where V is the tip of the pyramid. The slant edges VA, VB,VC and VD are all equal and the points C’

and D’ are the midpoints of VC and VD respectively. Calculate;

(i)The length of the slant edge. (5Marks)

(ii)The angle between the lines AV and VC (5 Marks)

(iii)The angle between planes ABCD and VBC (5 Marks)

(iv)The volume of the pyramid (5 Marks)