In the figure below, $ABC$ is an equilateral triangle of length $2k$ units while $ABD$ is a tr

Question

In the figure below, $ABC$ is an equilateral triangle of length $2k$ units while $ABD$ is a triangle where $AD$ is of length $k$ units and $\measuredangle CAD$ is $x$ radians. Show that $B{{D}^{2}}={{k}^{2}}\left( 5-2\cos x+2\sqrt{3}\sin x \right)$.[2] Given that $x$ is sufficiently small such that ${{x}^{3}}$ and higher powers of $x$ may be may be neglected, find an approximation in $x$ for the length $BD$, leaving your answers in exact form and in terms of $k$.[2]

Accepted Answer

(i) $BD^2=k^2(5-2\cos x+2\sqrt3\sin x)$; (ii) $BD\approx\sqrt3k+kx$

2022 HCI P1 Q2

2022 HCI P1 Q2

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