WAEC 2018 · Paper 2 · Q10

(a)
In $\triangle PQS$ , $|PQ| = 12\text{ cm}$ , $|PS| = 5\text{ cm}$ , $\angle SPQ = \angle PRQ = 90^\circ$ , where $R$ is on $SQ$ . Find, correct to three significant figures, $|PR|$ .
(b)
The lengths of two ladders, $L$ and $M$ , are $10\text{ m}$ and $12\text{ m}$ respectively. They are placed against a wall such that each ladder makes the same angle with the horizontal ground. If the foot of $L$ is $8\text{ m}$ from the foot of the wall, (i) draw a diagram to illustrate this information; (ii) calculate the height at which $M$ touches the wall.

Worked solution (try it first)

$\angle SPQ = 90^\circ$ , so $|SQ| = \sqrt{5^2 + 12^2} = 13$ cm.
$PR$ is perpendicular to $SQ$ , so it is the height of the triangle on the base $SQ$ .
Work out the area two ways: $\frac12 \times 5 \times 12 = \frac12 \times 13 \times |PR|$ .
So $|PR| = \frac{60}{13} \approx 4.62$ cm.

Draw the wall vertical and the two ladders leaning against it at the same angle to the ground, $L$ (10 m) with its foot 8 m from the wall and $M$ (12 m).

Ladder $L$ reaches $\sqrt{10^2 - 8^2} = 6$ m up the wall.
Both ladders make the same angle with the ground, so the two triangles are similar, and everything scales by $\frac{12}{10}$ .
So $M$ reaches $6 \times \frac{12}{10} = 7.2$ m up the wall.