# In a triangle ABC, sin A : sin B : sin C =1 : 2 : 3. Find the perimeter of the triangle if b = 4cm.

(a) 6 cm

(b) 24 cm

(c) 12 cm

(d) 8 cm

Last updated date: 24th Mar 2023

•

Total views: 307.8k

•

Views today: 2.84k

Answer

Verified

307.8k+ views

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.

Recently Updated Pages

If a spring has a period T and is cut into the n equal class 11 physics CBSE

A planet moves around the sun in nearly circular orbit class 11 physics CBSE

In any triangle AB2 BC4 CA3 and D is the midpoint of class 11 maths JEE_Main

In a Delta ABC 2asin dfracAB+C2 is equal to IIT Screening class 11 maths JEE_Main

If in aDelta ABCangle A 45circ angle C 60circ then class 11 maths JEE_Main

If in a triangle rmABC side a sqrt 3 + 1rmcm and angle class 11 maths JEE_Main

Trending doubts

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Epipetalous and syngenesious stamens occur in aSolanaceae class 11 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

A Short Paragraph on our Country India