WAEC 2016 · Paper 2 · Q5

(a)
In the diagram, $PQST$ is a parallelogram, $PQMR$ is a straight line, $|TS| = 8\text{ cm}$ , $|SM| = 6\text{ cm}$ ( $SM \perp PR$ ) and the area of triangle $PSR = 36\text{ cm}^2$ . Find the value of $|QR|$ .
(b)
A tree and a flagpole are on the same horizontal ground. A bird on top of the tree observes the top and bottom of the flagpole below it at angles of depression of $45^\circ$ and $60^\circ$ respectively. If the tree is $10.65\text{ m}$ high, calculate, correct to 3 significant figures, the height of the flagpole.

Worked solution (try it first)

Opposite sides of a parallelogram are equal, so $|PQ| = |TS| = 8$ cm.
$SM$ is the height of triangle $PSR$ on the base $PR$ : $\frac12 \times |PR| \times 6 = 36$ , so $|PR| = 12$ cm.
So $|QR| = 12 - 8 = 4\text{ cm}$ .

Put the bird at the top $B$ of the tree, 10.65 m up.
The angle of depression of the bottom of the flagpole is $60^\circ$ , so the horizontal distance from the tree to the flagpole is $d = \frac{10.65}{\tan 60^\circ}$
$\approx 6.149$ m.
Looking at the top of the flagpole at $45^\circ$ , the bird looks down through $d\tan 45^\circ = 6.149$ m.
The flagpole is the rest of the tree's height: $10.65 - 6.149 \approx 4.50\text{ m}$ .