Question

If the angles of a triangle are in the ratio $1:1:4$, then the ratio of the perimeter of the triangle to its largest side is:

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

Answer 1

The correct option is D

$\sqrt{3}+2:\sqrt{3}$

Explanation for the correct option.

Step 1: Find the angles

Given that, the angles of a triangle are in the ratio$1:1:4$.

Let $\mathrm{A},\mathrm{B},\mathrm{C}$ be the angles of the triangle.
&$A=x,B=x,C=4x$

Now using the angle sum property of triangle we have:
$\begin{array}{rcl}\mathrm{A}+\mathrm{B}+\mathrm{C}& =& 180°\\ x+x+4x& =& 180°\\ 6x& =& 180°\\ x& =& 30°\end{array}$
$\therefore \mathrm{B}=\mathrm{A}=30°\text{and}\mathrm{C}=4x=120°$

Step 2: Find the length of the sides

Now using sine rule, $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$
$$\begin{gathered} \therefore \frac{\mathrm{a}}{\sin 30°}=\frac{\mathrm{b}}{\sin 30°}=\frac{\mathrm{c}}{\sin 120°} \\ \therefore \frac{\mathrm{a}}{\frac{1}{2}}=\frac{\mathrm{b}}{\frac{1}{2}}=\frac{\mathrm{c}}{\frac{\sqrt{3}}{2}} \\ \therefore \mathrm{b}=\mathrm{a}\mathrm{and}\mathrm{c}=\sqrt{3}\mathrm{a} \end{gathered}$$

Step 3: Ratio of the perimeter of the triangle to its largest side

The perimeter of the triangle is given by the sum of the length of all sides.

$\begin{array}{rcl}P& =& \mathrm{a}+\mathrm{b}+\mathrm{c}\\ & =& \mathrm{a}+\mathrm{a}+\sqrt{3}\mathrm{a}\\ & =& (2+\sqrt{3})\mathrm{a}\end{array}$

Largest side$=\sqrt{3}a$

So, the ratio of the perimeter of the triangle to its largest side is:

$\begin{array}{rcl}\frac{\text{P erimeter}}{\text{Largest Side}}& =& \frac{(2+\sqrt{3})a}{\sqrt{3}a}\\ & =& \frac{2+\sqrt{3}}{\sqrt{3}}\end{array}$

Hence, option D is correct.