the diagram shows a sector OPQ of a circle with centre O. the radius of the circle is 18m and the angle POQ is 2π / 3 radians.

a) find the length of the arc PQ, giving the answer as a multiple of π

b) the tangents to the cirlce at the point P and Q meet at the point T. and the angles TPO AND TQO are both right angles

i) Angle PTQ = ∝ radians. find ∝ in terms of pi

ii)Find the area of the shaded region bounded by the arc PQ and the tangents TP and TQ

Guest Aug 16, 2019

#1**+1 **

the diagram shows a sector OPQ of a circle with centre O. the radius of the circle is 18m and the angle POQ is 2π / 3 radians.

a) find the length of the arc PQ, giving the answer as a multiple of π

b) the tangents to the cirlce at the point P and Q meet at the point T. and the angles TPO AND TQO are both right angles

i) Angle PTQ = ∝ radians. find ∝ in terms of pi

ii)Find the area of the shaded region bounded by the arc PQ and the tangents TP and TQ

a) arc length PQ = 18 * (2 pi)/3 = 36 pi / 3 = 12 pi

b) i) Angle PTQ will be supplemental to angle POQ = pi - (2pi)/3 = pi/3 = 60°

ii ) The area of the circular sector formed by arc PQ = (1/2)r^2 (2pi/3) = (1/2)(18)^2 (2pi)/3 = 108 pi m^2 (1)

[PTQO] will from a kite ...OQ = 18 and triangle TPO will form a 30-60-90 right triangle with QO = 18 opposite the 30° angle (angle QTO) and QT opposite the 60° angle (angle QOT)....so QT = 18√3

So the area of this kite = product of the legs of triangle OQT = QO * QT = 18 * 18√3 = 324√3 (2)

So...the area bounded by arc PQ and the tangents TP and TQ = (2) - (1) = 324√3 - 108 pi ≈ 221.9 m^2

CPhill Aug 16, 2019