(a)
In the diagram, AB // CD and BC // FE. \(\stackrel\frown{CDE} = 75^\circ\) and \(\stackrel\frown{DEF} = 26^\circ\). Find the angles marked x and y.
(b)
The diagram shows a circle ABCD with centre O and radius 7 cm. The reflex angle AOC = 190^\circ and < DAO = 35^\circ. Find :
(i) < ABC ; (ii) < ADC.
(c) Using the diagram in (b) above, calculate, correct to 3 significant figures, the length of : (i) arc ABC ; (ii) the chord AD. [Take \(\pi = 3.142\)].
Verified Correct Answer: .
Step-by-Step Mathematical Solution
(a) 105^\circ + 26^\circ + x = 180^\circ
131^\circ + x = 180^\circ
x = 180^\circ - 131^\circ
= 49^\circ.
131^\circ + x + 131^\circ + 360 - y = 360^\circ
131^\circ + 49^\circ + 131^\circ + 360 - y = 360^\circ
671^\circ - y = 360^\circ
y = 671^\circ - 360^\circ
= 311^\circ.
(b)(i) \(\stackrel\frown{ABC} = \frac{1}{2} \stackrel\frown{AOC} reflex\)
= \(\frac{1}{2} \times 190^\circ = 95^\circ\)
(ii) \(\stackrel\frown{ADC} = \frac{1}{2} \stackrel\frown{AOC}\)
= \(\frac{1}{2} (360^\circ - 190^\circ)\)
= \(\frac{1}{2} \times 170^\circ\)
= 85^\circ
(c) (i) \(\stackrel\frown{ABC} = \frac{360 - 190}{360} \times 2\pi r\)
= \(\frac{170}{360} \times 2 \times 3.142 \times 7\)
= \(20.77 cm\)
\(\approxeq 20.8 cm\) (3 significant figures)
(ii) \(AD = 2 \times 7\cos 35\)
= \(14 \times 0.8192\)
= \(11.468 cm \approxeq 11.5 cm\) (3 significant figures)
Source: Myschool Past Questions Archive (Record #52963)