(a) In < PQS, |PQ| = 12 cm, |PS| = 5 cm, < SPQ = < PRQ = 90°, Find, correct to three significant figures, |PR|.
(b) The length 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.