(a) In < PQS, |PQ| = 12 cm, |PS| = 5 cm, < SPQ = < PRQ = 90°, Find, correct to three significant figures, |PR|.
(b) The lengths of two ladders, L and M are 10m and 12m respectively. They are placed against a wall such that each ladder makes angle with the horizontal ground. If the foot of L is 8m from the foot of the wall.
(i) Draw a diagram to illustrate this information; (ii) Calculate the height at which M touches the wall.
Explanation
(a) 
In triangle SPQ, \(|SQ|^{2} = 5^{2} + 12^{2}\) (Pythagoras theorem)
= \(25 + 144 = 169\)
\(|SQ| = \sqrt{169} = 13 cm\)
Angle b is common to triangles SPQ and PRS are similar.
Using \(\sin b = \frac{12}{|SQ|} = \frac{|PR|}{5}\)
\(\sin b = \frac{12}{13} = \frac{|PR|}{5}\)
\(|PR| = \frac{12 \times 5}{13} \approxeq 4.62 cm\) (to 3 s.f)
(b)(i) 
\(h^{2} = 10^{2} - 8^{2} = 36\)
\(h = \sqrt{36} = 6 cm\)
(ii) In the smaller triangle, \(\cos x = \frac{8}{10} = 0.8\)
\(\cos^{-1} (0.8) = 36.87°\)
Since these are corresponding angles, x = x in the bigger triangle.
\(\sin x = \frac{y}{12}\)
\(y = 12 \sin x = 12 \sin 36.87\)
= \(12 \times 0.6\)
= 7.20 m
The ladder M touches the wall at a height 7.2 m above the horizontal ground.