Question

Two sides of a triangle are given by the roots of the equations $x^2-5x+6=0$ and the angle between the sides is $\frac{\pi }{3}$ Then, the perimeter of the triangle is

• A. $5+\sqrt{2}$
• B. $5+\sqrt{3}$
• C. $5+\sqrt{5}$
• D. $5+\sqrt{7}$

Answer 1

The correct option is D

$5+\sqrt{7}$

The explanation for the correct answer.

Step 1. Find the length of the two side

Given equation is $x^2-5x+6=0$

$$\begin{gathered} \therefore x^2–5x+6=0 \\ \Rightarrow (x–2)(x–3)=0 \end{gathered}$$

So $x=2$ and $x=3$

Let $b=2$ and $c=3$

Step 2. Find the length of the third side

The angle between the sides $b$and $c$ is $A=\frac{\mathrm{\pi }}{3}$

We know $\mathbf{cos}\mathbf{}\mathbf{A}\mathbf{}\mathbf{=}\frac{\left(b^2+c^2–a^2\right)}{\mathbf{2}\mathbf{bc}}$

$\Rightarrow \cos {60}^{\circ }=\frac{\left(2^2+3^2-a^2\right)}{2\times 2\times 2}$

$$\begin{gathered} \Rightarrow \frac{1}{2}=\frac{(13–a^2)}{12} \\ \Rightarrow 6=13–a^2 \\ \Rightarrow a^2=7 \end{gathered}$$

So, $a=\sqrt{7}$

Step 3. Find the perimeter of the triangle

Perimeter $=a+b+c$

$$\begin{gathered} =\sqrt{7}+2+3 \\ =5+\sqrt{7} \end{gathered}$$

Hence, option(D) is the correct .