Problem Statement
In the diagram, the equation of the line $AD$ is $y = \sqrt{3}(x-1)$. $BD$ bisects $\angle ADC$. If the coordinates of $B$ are $(p, q)$, what is the value of $q$?
![Problem Diagram]()
Problem Link
Solution
$$f(x) = \sqrt{3}(x-1)$$ $$\text{Since D lies on the x-axis, } \implies D(x, 0) \text{ and } 0 = \sqrt{3}(x-1) \implies x = 1 \implies D(1, 0)$$ $$\text{Since A lies on the line } x = 13, \implies A(13, f(13)) \implies A(13, 12\sqrt{3})$$ $$\therefore |AC| = 12\sqrt{3}, \; |DC| = 12$$ $$\therefore \triangle ADC \text{ is a 30-60-90 triangle}$$ $$\text{Since } BD \text{ bisects } \angle ADC \implies \angle ADB = \angle BDC = 30^{\circ} \implies \angle DBC = 60^{\circ}$$ $$\therefore \triangle DBC \text{ is a 30-60-90 triangle and } \triangle DBC \sim \triangle ADC$$ $$\therefore \dfrac{|BC|}{|DC|} = \dfrac{|DC|}{|AC|} $$ $$\therefore \dfrac{|BC|}{12} = \dfrac{12}{12\sqrt{3}} $$ $$\therefore |BC| = \dfrac{12}{\sqrt{3}}$$ $$\therefore B(13, \dfrac{12}{\sqrt{3}}) \implies q = \dfrac{12}{\sqrt{3}}$$ 
