Answer

Verified

457.5k+ views

Hint: Use area of triangle formula, where two sides of triangle and angle between them is given. Use sine rule related with circumradius to get the given relation.

Complete step-by-step answer:

Here, we have given R as a radius of circumcircle i.e. circumradius and need to prove the relation;

$R=\dfrac{abc}{4S}$……………….(1)

Where (a, b, c) are sides of the triangle as denoted in the diagram.

Where O is the centre of the circle C, which is circumscribing the triangle ABC.

R = Circumradius of triangle ABC.

We can write sine rule in $\Delta ABC$ involving circumradius R as;

$\dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}=\dfrac{1}{2R}..............\left( 2 \right)$

As we have a formula of area with involvement of two sides and angle between them.

Let the area be represented by S.

$Area=S=\dfrac{1}{2}bc\sin A=\dfrac{1}{2}ab\sin C=\dfrac{1}{2}ac\sin B........\left( 3 \right)$

Now, from equation (2) and (3), we can write an equation with respect to one angle as

$\dfrac{\sin A}{a}=\dfrac{1}{2R}\text{ and }S=\dfrac{1}{2}bc\sin A$

Substituting value of sin A from the relation $\dfrac{\sin A}{a}=\dfrac{1}{2R}\text{ to }S=\dfrac{1}{2}bc\sin A$, we get;

As $\sin A=\dfrac{a}{2R}$ from the first relation, now putting value of sin A in $S=\dfrac{1}{2}bc\sin A$, we get

$\begin{align}

& S=\dfrac{1}{2}bc\dfrac{a}{2R} \\

& S=\dfrac{abc}{4R} \\

\end{align}$

Transferring R to other side, we get;

$R=\dfrac{abc}{4S}$

Hence, the relation given in the problem is true.

Note: One can go wrong with the formula of area of the triangle. One can apply heron’s formula for proving i.e.

$\begin{align}

& S=\sqrt{s\left( s-a \right)\left( s-b \right)\left( s-c \right)} \\

& s=\dfrac{a+b+c}{2} \\

\end{align}$

Which will make the solution very complex.

One can go wrong while writing sine rule as;

$\dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}=\dfrac{2R}{1}$ which is wrong.

Correct equation of sine rule will be,

$\dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}=\dfrac{1}{2R}$.

Complete step-by-step answer:

Here, we have given R as a radius of circumcircle i.e. circumradius and need to prove the relation;

$R=\dfrac{abc}{4S}$……………….(1)

Where (a, b, c) are sides of the triangle as denoted in the diagram.

Where O is the centre of the circle C, which is circumscribing the triangle ABC.

R = Circumradius of triangle ABC.

We can write sine rule in $\Delta ABC$ involving circumradius R as;

$\dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}=\dfrac{1}{2R}..............\left( 2 \right)$

As we have a formula of area with involvement of two sides and angle between them.

Let the area be represented by S.

$Area=S=\dfrac{1}{2}bc\sin A=\dfrac{1}{2}ab\sin C=\dfrac{1}{2}ac\sin B........\left( 3 \right)$

Now, from equation (2) and (3), we can write an equation with respect to one angle as

$\dfrac{\sin A}{a}=\dfrac{1}{2R}\text{ and }S=\dfrac{1}{2}bc\sin A$

Substituting value of sin A from the relation $\dfrac{\sin A}{a}=\dfrac{1}{2R}\text{ to }S=\dfrac{1}{2}bc\sin A$, we get;

As $\sin A=\dfrac{a}{2R}$ from the first relation, now putting value of sin A in $S=\dfrac{1}{2}bc\sin A$, we get

$\begin{align}

& S=\dfrac{1}{2}bc\dfrac{a}{2R} \\

& S=\dfrac{abc}{4R} \\

\end{align}$

Transferring R to other side, we get;

$R=\dfrac{abc}{4S}$

Hence, the relation given in the problem is true.

Note: One can go wrong with the formula of area of the triangle. One can apply heron’s formula for proving i.e.

$\begin{align}

& S=\sqrt{s\left( s-a \right)\left( s-b \right)\left( s-c \right)} \\

& s=\dfrac{a+b+c}{2} \\

\end{align}$

Which will make the solution very complex.

One can go wrong while writing sine rule as;

$\dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}=\dfrac{2R}{1}$ which is wrong.

Correct equation of sine rule will be,

$\dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}=\dfrac{1}{2R}$.

Recently Updated Pages

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

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

Advantages and disadvantages of science

10 examples of friction in our daily life

Trending doubts

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Which are the Top 10 Largest Countries of the World?

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

10 examples of law on inertia in our daily life

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

In 1946 the Interim Government was formed under a Sardar class 11 sst CBSE

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