(a) Without using calculator or mathematical tables, evaluate (frac{3}{sqrt{3}}(frac{2}{sqrt{3}} – frac{sqrt{12}}{6}))
(b) In the diagram, O is the centre of the circle. The side AB is produced to E, < ACB = 49° and < CBE = 68°. Calculate,
(i) the interior angle AOC ; (ii) < BOC.
Explanation
(a) (frac{3}{sqrt{3}}(frac{2}{sqrt{3}} – frac{sqrt{12}}{6}))
= (frac{6}{sqrt{9}} – frac{3sqrt{12}}{6sqrt{3}})
= (2 – frac{2sqrt{3}}{2sqrt{3}})
= (2 – 1 = 1)
(b)(i) (< ABC = 180° – 68° = 112°)
(< AOC = 2 times 112° = 224°)
(hat{O} = < AOC (obtuse))
= (360° – 224° = 136°)
(ii) In (Delta AOC), OA = OC (radii)
(therefore Delta AOC = isosceles)
(hat{A} = < OAC ; hat{C} = < ACO ; hat{A} = hat{C} = x)
(hat{O} + hat{A} + hat{C} = 180°)
(136° + x + x = 180° implies 2x = 44°)
(x = 22°)
In (Delta BOC), OB = OC (radii)
(therefore Delta BOC = isosceles)
(< OCB = 49° + 22°= 71°)
(< OBC = < OCB = 71°)
(< OBC + < OCB + < BOC = 180°)
(71° + 71° + < BOC = 180°)
(< BOC = 180° – 142° = 38°)