PT3 Mathematics 2017, Question 6

Question 6 (a):
In Diagram 6.1, PWVU is a straight line.

(i) Name the corresponding angle of ∠SWV.
(ii) State the value of ∠RVW.
(iii) Diagram 6.2 shows two parallel lines.


Based on Diagram 6.2, state two pairs of alternate angles.

Solution:
(i) ∠TVU

(ii) 180^o – 40^o – 90^o = 50^o

(iii)
b and e
c and d

Question 6 (b):
Diagram 6.3 shows a lighthouse at the top of a cliff and two ships docked near the cliff.

Point B, C and D are aligned. Point A is vertically above point B. The distance between points C and titik D is 80 m.
Calculate the value of θ.

Solution:
$\begin{array}{l}A{B}^2+{120}^2={130}^2\\ A{B}^2={130}^2-{120}^2\\ AB=\sqrt{{130}^2-{120}^2}\\ AB=\sqrt{2500}\\ AB=50\text{ m}\\ \\ BC=120-80=40\text{ m}\\ \tan \theta =\frac{AB}{BC}\\ \tan \theta =\frac{50}{40}\\ \theta ={51}^{o}20‘\end{array}$

Question 6 (c):
Diagram 6.4 shows a right angled triangle card. A circle with radius t cm has been cut from the card.


(i) Express the area of the remaining card, A, in terms of s and t.
(ii) Hence, calculate the value of A when s = 5 cm and t = 7 cm.

Solution:
(i)
$\begin{array}{l}A=\left(\frac{1}{\cancel{2}}\times 3s\times {\cancel{4}}^{2}t\right)-\pi t^2\\ A=6st-\pi t^2\end{array}$

(ii)
$\begin{array}{l}A=6st-\pi t^2\\ A=6\left(5\right)\left(7\right)-\left(\frac{22}{7}\times 7^2\right)\\ A=210-154\\ A=56{\text{ cm}}^2\end{array}$