Exam

Olympiad workbook

Subject

IOQM

Chapter

Properties of Triangles

Olympiad workbook · IOQM · Properties of Triangles

Let x, y, z be positive real number such that x^2+y^2=49, y^2+y...

Let $x, y, z$ be positive real number such that $x^2+y^2=49, y^2+y z+z^2=36$ and $x^2+\sqrt{3} x z+z^2=25$. If the value of $2 x y+\sqrt{3} y z+z x$ can be written as $p \sqrt{q}$ where $p, q$ are integers and $q$ is not divisible by square of any prime number, find $\mathrm{p}+\mathrm{q}$.

Step-by-step solution

$$\begin{aligned} & x^2+y^2=49 \\ & \frac{x^2+y^2-49}{2 x y}=0 \\ & \cos A=0, A=90^{\circ} \\ & \sin \text { in } \cos B=\frac{y^2+z^2-36}{2 y z}=-\frac{1}{2}, B=120^{\circ} \\ & \cos C=\frac{z^2+x^2-25}{2 x z}=-\frac{\sqrt{3}}{2}, C=150^{\circ} \end{aligned}$$
$$\begin{aligned} & \text { Area of } \mathrm{PQR}=\sqrt{9 \times 4 \times 3 \times 2}=6 \sqrt{6} \\ & 2 \mathrm{xy}+\sqrt{3} \mathrm{yz}+\mathrm{zx}=4 \text { Area of } \mathrm{PQR} \\ & =4 \times 6 \sqrt{6}=24 \sqrt{6} \\ & 24+6=30\end{aligned}$$