Question

In a triangle$ABC$, $\sin A:\sin B:\sin C=1:2:3$ if $b=4cm$, the perimeter of the triangle is,
A. $6cm$
B. $24cm$
C. $12cm$
D. $8cm$

Answer 1

Hint: To solve this question, we will use the Law of sine $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$. We will assume the value of the angles from the given ratio $\sin A:\sin B:\sin C=1:2:3$. Then will first take $\frac{a}{\sin A}=\frac{b}{\sin B}$ and substitute the given values of side $b=4cm$and angles $\sin A,\sin B$and determine the value of side $a$. After this we will take $\frac{b}{\sin B}=\frac{c}{\sin C}$ and substitute the values of side $b=4cm$ and angles $\sin B,\sin C$ and determine the value of side $c$. We will then substitute the values of all the sides of the triangle in the formula of the perimeter and determine its value.
Formula used:
$P=a+b+c$, where $a,b,c$are the sides.
Law of sine :
$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$
Complete step-by-step solution:
We are given a triangle$ABC$ in which the ratio of all the three angles are $\sin A:\sin B:\sin C=1:2:3$ and the value of one side is $b=4cm $and we are required to calculate the perimeter .
Let the values of the angles from the ratio be $\sin A=x,\,\sin B=2x$ and $\sin C=3x$.
Now first we will determine the value of the sides $a$ and $c$.
Using the Law of sine $\frac{a}{\sin A}=\frac{b}{\sin B}$, we will first determine the value of side $a$.
$\begin{align}
  & \frac{a}{x}=\frac{4}{2x} \\
 & a=2cm \\
\end{align}$
Now we will determine the value of the side $c$ with the help of Law of sine $\frac{b}{\sin B}=\frac{c}{\sin C}$.
$\begin{align}
  & \frac{4}{2x}=\frac{c}{3x} \\
 & c=6cm \\
\end{align}$
We will now calculate the perimeter of the triangle $ABC$,
$P=a+b+c$.
$\begin{align}
  & P=2+4+6 \\
 & =12cm
\end{align}$
The perimeter of the triangle$ABC$ is $12cm$when the ratio of the angles is $\sin A:\sin B:\sin C=1:2:3$ and the value of one of the sides is $b=4cm$. Hence the correct option is (C).
Note:
Perimeter can be defined as the sum of the lengths of all the boundaries of a polygon. The smallest polygon is a triangle with three sides so its perimeter will be the sum of all its three sides. There are different formulas of perimeter for different types of triangles like equilateral, isosceles and scalene but the most basic method to find perimeter is add the lengths of all the sides of a polygon.