(a) The ratio of the interior angle to the exterior angle of a regular polygon is 5 : 2, Find the number of sides of the polygon.
(b)
The diagram shows a circle PQRS with centre O, < UQR = 68°, < TPS = 74° and < QSR = 40°. Calculate the value of < PRS.
Explanation
(a) Let the diagram below represent a section of the polygon
Also, let :
(i) represent the size of an interior angle; (e) represent the size of an exterior angle.
Then (frac{i}{e} = frac{5}{2})
(implies i = frac{5}{2}e)
But (i + e = 180°) (sum of angles on a straight line)
Substitute (frac{5}{2}e) for (i) in the equation
(frac{5}{2}e + e = 180°)
(frac{7}{2}e = 180°)
(e = frac{180° times 2}{7} = frac{360°}{7})
Number of sides of the polygon = (frac{360°}{text{size of one exterior angle}})
= (360° div frac{360°}{7})
(360° times frac{7}{360°} = 7 sides).
(b)
In the diagram above, < PSR = 68° (interior angle of a cyclic quad = opp exterior angle)
< PSQ = 68° – 40° = 28°
< PRQ = 28° (angles in same segment)
< SRQ = 74° (interior angle of a cyclic quad = opp exterior angle)
hence, < PRS = 74° – 28° = 46°