Answer

Verified

450.3k+ views

Let us consider a triangle$\Delta abc$, where $a,b,c$ are the sides of triangle

Given $c$ is the length of the third side of the triangle, then $a,b$ be the other two sides of the triangle.

Given sum of lengths of two sides = $x$

$a + b = x \to (1)$

And also given product of length of same two sides = $y$

$ab = y \to (2)$

Given condition

$ \Rightarrow $${x^2} - {y^2} = {c^2}$

Let us substitute the $x$ and $y$ values in the above equation

$

\Rightarrow {(a + b)^2} - {c^2} = ab \\

\Rightarrow {a^2} + {b^2} + 2ab - {c^2} = ab \\

\Rightarrow {a^2} + {b^2} - {c^2} = ab - 2ab \\

\Rightarrow {a^2} + {b^2} - {c^2} = - ab \\

$

Let us divide with $2ab$on both sides we get

$

\Rightarrow \dfrac{{{a^2} + {b^2} - {c^2}}}{{2ab}} = \dfrac{{ - ab}}{{2ab}} \\

\Rightarrow \dfrac{{{a^2} + {b^2} - {c^2}}}{{2ab}} = \dfrac{{ - 1}}{2} \to (3) \\

$

Apply the cosine rule formula where $COSC = \dfrac{{{a^2} + {b^2} - c2}}{{2ab}}$

From the cosine rule we can rewrite the equation $'3'as$

$

COSC = \dfrac{{ - 1}}{2} \\

\angle C = \dfrac{{2\pi }}{3} \\

$

We know that circumradius of triangle is $R = \dfrac{C}{{2SINC}}$

On substituting the value we get $R = \dfrac{C}{{\sqrt 3 }}$

Therefore circumradius of triangle is $R = \dfrac{C}{{\sqrt 3 }}$

( B) is the correct option

NOTE: Make a note that after substituting the value in the given condition we have divided the equation with $2ab$ on both sides, where we directly get the required value of circumradius after applying cosine rule.

Given $c$ is the length of the third side of the triangle, then $a,b$ be the other two sides of the triangle.

Given sum of lengths of two sides = $x$

$a + b = x \to (1)$

And also given product of length of same two sides = $y$

$ab = y \to (2)$

Given condition

$ \Rightarrow $${x^2} - {y^2} = {c^2}$

Let us substitute the $x$ and $y$ values in the above equation

$

\Rightarrow {(a + b)^2} - {c^2} = ab \\

\Rightarrow {a^2} + {b^2} + 2ab - {c^2} = ab \\

\Rightarrow {a^2} + {b^2} - {c^2} = ab - 2ab \\

\Rightarrow {a^2} + {b^2} - {c^2} = - ab \\

$

Let us divide with $2ab$on both sides we get

$

\Rightarrow \dfrac{{{a^2} + {b^2} - {c^2}}}{{2ab}} = \dfrac{{ - ab}}{{2ab}} \\

\Rightarrow \dfrac{{{a^2} + {b^2} - {c^2}}}{{2ab}} = \dfrac{{ - 1}}{2} \to (3) \\

$

Apply the cosine rule formula where $COSC = \dfrac{{{a^2} + {b^2} - c2}}{{2ab}}$

From the cosine rule we can rewrite the equation $'3'as$

$

COSC = \dfrac{{ - 1}}{2} \\

\angle C = \dfrac{{2\pi }}{3} \\

$

We know that circumradius of triangle is $R = \dfrac{C}{{2SINC}}$

On substituting the value we get $R = \dfrac{C}{{\sqrt 3 }}$

Therefore circumradius of triangle is $R = \dfrac{C}{{\sqrt 3 }}$

( B) is the correct option

NOTE: Make a note that after substituting the value in the given condition we have divided the equation with $2ab$ on both sides, where we directly get the required value of circumradius after applying cosine rule.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Why Are Noble Gases NonReactive class 11 chemistry CBSE

Let X and Y be the sets of all positive divisors of class 11 maths CBSE

Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE

Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE

Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

How many crores make 10 million class 7 maths CBSE

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

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

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

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths