Theory question 111
Angles and Polygons(a) In the diagram, AOB is a straight line, ∠AOC = 3(x + y)°, ∠COB = 45°, ∠AOD = (5x + y)° and ∠DOB = y°. Find the values of x and y.
(b) From two points on opposite sides of a pole 33 m high, the angles of elevation of the top of the pole are 53° and 67°. If the two points and the base of the pole are on the same horizontal level, calculate, correct to three significant figures, the distance between the two points.
Show the model answer
(a) Angles on a straight line: 3(x + y) + 45 = 180, so x + y = 45.
(5x + y) + y = 180, so 5x + 2y = 180.
Substituting x = 45 − y: 225 − 3y = 180, so y = 15 and x = 30.
(b) Distances from the pole: 33/tan 53° ≈ 24.87 m and 33/tan 67° ≈ 14.01 m.
Distance between the points ≈ 24.87 + 14.01 ≈ 38.9 m.
The points are on opposite sides, so add the two horizontal distances.