The diagram below shows $\Delta ABC$ and $\Delta AED$. $\Delta ABC$ is an equilateral triangle

Question

The diagram below shows $\Delta ABC$ and $\Delta AED$. $\Delta ABC$ is an equilateral triangle with each side measuring $2$ units and $\Delta AED$ is a right-angled triangle with $\angle ADE = \frac{\pi}{4} + x$. Points $C$ and $E$ both lie on the line $BD$. Show that $BD = 1 + \frac{\sqrt{3}}{\tan\left(\frac{\pi}{4} + x\right)}$.[2] Given that $x$ is sufficiently small for $x^3$ and higher powers of $x$ to be neglected, show that $BD \approx a + bx + cx^2$ where $a$, $b$ and $c$ are exact constants to be determined.[4]

Accepted Answer

(b) $a=1+\sqrt3,\ b=-2\sqrt3,\ c=2\sqrt3$

2025 SAJC P2 Q3

2025 SAJC P2 Q3

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