Exam

Olympiad workbook

Subject

IOQM

Chapter

Properties of Triangles

Olympiad workbook · IOQM · Properties of Triangles

In a triangle A B C, B A C=90^ . Let D be the point on B C such...

In a triangle $A B C, \angle B A C=90^{\circ}$. Let $D$ be the point on $B C$ such that $A B+B D=A C+C D$. Suppose $B D: D C=2: 1$. If $\frac{A C}{A B}=\frac{m+\sqrt{p}}{n}$, where $m, n$ are relatively prime positive integers and $p$ is a prime number, determine the value of $m+n+p$.

Step-by-step solution

$A B+B D=A C+C D$
$$\begin{aligned} & y+2 x=A C+x \\ & \Rightarrow \quad A C=x+y \\ & \Rightarrow \quad(x+y)^2+y^2=(3 x)^2 \\ & \Rightarrow \quad x^2+2 y^2+2 x y=9 x^2 \\ & \Rightarrow \quad 8 x^2-2 y^2-2 x y=0 \\ & \Rightarrow \quad 4 a^2-a-1=0 \\ & {\left[\begin{array}{ll}\therefore & a=\frac{x}{y}\end{array}\right]} \\ & \Rightarrow \quad a=\frac{1 \pm \sqrt{1+16}}{8} \\ & \Rightarrow \quad a=\frac{1 \pm \sqrt{17}}{8} \\ & \Rightarrow \quad \frac{A C}{A B}=\frac{x}{y}+1=a+1 \\ & \Rightarrow \quad \frac{A C}{A B}=\frac{1+\sqrt{17}}{8}+1 \\ & \Rightarrow \quad \frac{A C}{A B}=\frac{9+\sqrt{17}}{8} \\ & \Rightarrow \quad m+n+p=34\end{aligned}$$