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Nernst Equation

In Δ ABC ...

Question

In $Δ$ $A B C$ the sides opposite to angles $A, B, C$ are denoted by $a, b, c$ respectively. Let $r_{1}, r_{2}$ and $r_{3}$ be the ex-radii of the triangle.
If $r_{1} = 2 r_{2} = 3 r_{3}$ , then the perimeter of the triangle is equal to?

3 a

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3 b

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2 c

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3 (a + b + c)

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Solution

The correct option is B $2 c$
We have, $△ A B C, r_{1} = 2 r_{2} = 3 r_{3}$
$\Rightarrow \frac{Δ}{s - a} = \frac{2 Δ}{s - b} = \frac{3 Δ}{s - c} = \frac{1}{k}$ ,
$\Rightarrow s = a + k Δ$ ......(1),
$s = b + 2 k Δ$ .....(2)
and $s = c + 3 k Δ$ ........(3)
add all above equations
$\Rightarrow 3 s = 2 s + 6 k Δ$
$\Rightarrow s = 6 k Δ$
$\Rightarrow k Δ = \frac{s}{6}$
Now using (3) $s = c + \frac{s}{2}$
$\Rightarrow s = 2 c$

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Similar questions

Δ

A B C

the sides opposite to angles

A, B, C

are denoted by

a, b, c

respectively.

r_{1}, r_{2}, r_{3}

are the ex-radii then if

r_{1} = 2 r_{2} = 3 r_{3}

, the value of

a : b

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Δ

A B C

the sides opposite to angles

A, B, C

are denoted by

a, b, c

respectively. let

r_{1}, r_{2}

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Δ

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the sides opposite to angles

A, B, C

are denoted by

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respectively. Let

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and

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be the ex-radii of the traingle and

r

be the in-radius.

Find the value of

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Δ

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the sides opposite to angles

A, B, C

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In Δ ABC the sides opposite to angles A,B,C are denoted by a,b,c respectively. Let r1,r2 and r3 be the ex-radii of the triangle.If r1=2r2=3r3, then the perimeter of the triangle is equal to?

The correct option is B 2cWe have, △ABC,r1=2r2=3r3⇒Δs−a=2Δs−b=3Δs−c=1k,⇒s=a+kΔ......(1), s=b+2kΔ.....(2) and s=c+3kΔ........(3)add all above equations⇒3s=2s+6kΔ⇒s=6kΔ⇒kΔ=s6Now using (3) s=c+s2 ⇒s=2c

In $Δ$ $A B C$ the sides opposite to angles $A, B, C$ are denoted by $a, b, c$ respectively. Let $r_{1}, r_{2}$ and $r_{3}$ be the ex-radii of the triangle.
If $r_{1} = 2 r_{2} = 3 r_{3}$ , then the perimeter of the triangle is equal to?