(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 1...
Question 1 Report
(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.
Answer Details
(a)
In right-angled triangle \(SPQ\),
\[SQ^2=PS^2+PQ^2=5^2+12^2=169.\]
Hence, \(SQ=13\text{ cm}\).
\(PR\) is the perpendicular height from \(P\) to the hypotenuse \(SQ\). Equating the two expressions for the area of \(\triangle SPQ\):
\[\frac12(5)(12)=\frac12(13)(PR).\]
\[PR=\frac{5\times12}{13}=4.615\ldots\text{ cm}.\]
\[\boxed{PR=4.62\text{ cm}}\qquad\text{(correct to 3 significant figures).}\]
(b)(i) The required diagram is:
(b)(ii)
For ladder \(L\), let \(\theta\) be its angle with the horizontal ground. Then
\[\cos\theta=\frac{8}{10}=0.8.\]
Therefore,
\[\sin\theta=\sqrt{1-0.8^2}=\sqrt{0.36}=0.6.\]
Since ladder \(M\) makes the same angle \(\theta\) with the ground, if \(h\) is the height at which it touches the wall,
\[\sin\theta=\frac{h}{12}.\]
\[h=12(0.6)=7.2\text{ m}.\]
\[\boxed{\text{Ladder }M\text{ touches the wall }7.2\text{ m above the ground.}}\]
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