Exam

Olympiad workbook

Subject

IOQM

Chapter

Straight Lines

Olympiad workbook · IOQM · Straight Lines

In a rectangle A B C D, E is the midpoint of A B: F is point on A C...

In a rectangle $A B C D, E$ is the midpoint of $A B: F$ is point on $A C$ such that $B F$ is perpendicular to $A C$ and $F E$ perpendicular to $B D$. Suppose $B C=8 \sqrt{3}$. Find $A B$.

Step-by-step solution

$\sin \theta=\frac{8 \sqrt{3}}{2 x}=\frac{4 \sqrt{3}}{x}$
$$\begin{aligned} & F A=2 x \cos \theta \\ & F B=2 x \sin \theta \\ & M B=F B x \sin 2 \theta=2 x \sin \theta \sin 2 \theta \\ & M E^2+M B^2=B E^2=x^2 \\ & x^2 \sin ^2 \theta+4 x^2 \sin ^2 \theta \sin ^2 2 \theta=x^2 \\ & \sin ^2 \theta+4 \sin ^2 \theta \sin ^2 2 \theta=1 \\ & 4 \sin ^2 \theta \sin ^2 2 \theta=\cos ^2 \theta \\ & 16 \sin ^4 \theta=1 \\ & \sin ^2 \theta=\frac{1}{4} \quad \Rightarrow \quad \sin \theta=\frac{1}{2} \\ & \frac{4 \sqrt{3}}{x}=\frac{1}{\sqrt{3}} \quad \Rightarrow \quad x=12 \Rightarrow 2 x=A B=24 \text { Ans. } \end{aligned}$$