Answer

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Hint: To find the perimeter, we make use of sine rule, to find the values of a and c. Then we can calculate the perimeter of the triangle by adding a, b and c.

Complete step-by-step answer:

To explain further now, we will refer to the triangle above. The sides a, b and c represent the sides opposite angles A, B and C (as shown in the triangle above). In the question, we are already given that b = 4cm. Thus, all we have to do is calculate the value of a and c.

To do so, we use the sine rule. By sine rule, we have,

$\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}=k$

Thus, we have,

a=k$\sin A$ -- (1)

b=k$\sin B$-- (2)

c=k $\sin C$-- (3)

From the question, we also have,

$\sin A:\sin B:\sin C$=1:2:3

Thus, $\dfrac{\sin A}{\sin B}=\dfrac{1}{2}$

Also,

$\begin{align}

& \dfrac{\sin B}{\sin C}=\dfrac{2}{3} \\

& \dfrac{\sin C}{\sin B}=\dfrac{3}{2} \\

\end{align}$

We will use these results in the below calculations-

Now, we divide (1) by (2), we get,

$\begin{align}

& \dfrac{a}{b}=\dfrac{\sin A}{\sin B} \\

& a=b\times \dfrac{\sin A}{\sin B} \\

& a=4\times \dfrac{1}{2} \\

& a=2 \\

\end{align}$

Now, we divide (2) by (3), we get,

$\begin{align}

& \dfrac{b}{c}=\dfrac{\sin B}{\sin C} \\

& c=b\times \dfrac{\sin C}{\sin B} \\

& c=4\times \dfrac{3}{2} \\

& c=6 \\

\end{align}$

Now, since we have the values of a, b and c. We are now ready to calculate the perimeter of the triangle. Thus,

Perimeter = a+b+c

Perimeter = 2+4+6

Perimeter = 12 cm

Hence, the correct answer is (c) 12 cm.

Note: Although not mentioned in the question, as a general rule, it is important to remember that a, b and c are sides to the angles A, B and C respectively. Further, to get the intuition of using sine rule, one can get a hint of this from the sine ratios of various angles given in the question. Whenever such a ratio is given, it is always better to use the sine rule to solve the problem.

Complete step-by-step answer:

To explain further now, we will refer to the triangle above. The sides a, b and c represent the sides opposite angles A, B and C (as shown in the triangle above). In the question, we are already given that b = 4cm. Thus, all we have to do is calculate the value of a and c.

To do so, we use the sine rule. By sine rule, we have,

$\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}=k$

Thus, we have,

a=k$\sin A$ -- (1)

b=k$\sin B$-- (2)

c=k $\sin C$-- (3)

From the question, we also have,

$\sin A:\sin B:\sin C$=1:2:3

Thus, $\dfrac{\sin A}{\sin B}=\dfrac{1}{2}$

Also,

$\begin{align}

& \dfrac{\sin B}{\sin C}=\dfrac{2}{3} \\

& \dfrac{\sin C}{\sin B}=\dfrac{3}{2} \\

\end{align}$

We will use these results in the below calculations-

Now, we divide (1) by (2), we get,

$\begin{align}

& \dfrac{a}{b}=\dfrac{\sin A}{\sin B} \\

& a=b\times \dfrac{\sin A}{\sin B} \\

& a=4\times \dfrac{1}{2} \\

& a=2 \\

\end{align}$

Now, we divide (2) by (3), we get,

$\begin{align}

& \dfrac{b}{c}=\dfrac{\sin B}{\sin C} \\

& c=b\times \dfrac{\sin C}{\sin B} \\

& c=4\times \dfrac{3}{2} \\

& c=6 \\

\end{align}$

Now, since we have the values of a, b and c. We are now ready to calculate the perimeter of the triangle. Thus,

Perimeter = a+b+c

Perimeter = 2+4+6

Perimeter = 12 cm

Hence, the correct answer is (c) 12 cm.

Note: Although not mentioned in the question, as a general rule, it is important to remember that a, b and c are sides to the angles A, B and C respectively. Further, to get the intuition of using sine rule, one can get a hint of this from the sine ratios of various angles given in the question. Whenever such a ratio is given, it is always better to use the sine rule to solve the problem.

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