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Thermal Expansion

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 traingle and $r$ be the in-radius.
Find the value of $r_{1} + r_{2} + r_{3} - r$

4 R

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2 R

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R

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3 R

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Solution

The correct option is D $4 R$
We know, $r_{1} + r_{2} + r_{3} - r = \frac{Δ}{s - a} + \frac{Δ}{s - b} + \frac{Δ}{s - c} - \frac{Δ}{s}$
$= Δ [\frac{s - b + s - a}{(s - a) (s - b)} + \frac{(s - s + c)}{s (s - c)}]$
$= Δ [\frac{c}{(s - a) (s - b)} + \frac{c}{s (s - c)}]$
$= Δ c [\frac{s (s - c) + (s - a) (s - b)}{s (s - a) (s - b) (s - c)}]$
$= \frac{Δ c}{Δ^{2}} [2 s^{2} - s (a + b + c) + a b]$
$= (\frac{c}{Δ}) [2 s^{2} - s (2 s) + a b] = 4 (\frac{a b c}{4 Δ}) = 4 R$

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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. let

r_{1}, r_{2}

and

r_{3}

be the ex-radii of the triangle.

The value of

\frac{r_{1}}{b c} + \frac{r_{2}}{c a} + \frac{r_{3}}{a b}

is equal to?

Δ

A B C

the sides opposite to angles

A, B, C

are denoted by

a, b, c

respectively, then

r_{1}^{2} + r_{2}^{2} + r_{3}^{3} + r^{2} = ?

(where

r

=

in-radius, R

=

circumradius,

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

are ex-radii)

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

is equal to?

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.

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

, then the perimeter of the triangle is equal to?

Q. Assertion :In a

△ A B C

, if

a < b < c

and

r

is inradius and

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

are the axradii opposite to angle

A, B, C

respectively, then

r < r_{1} < r_{2} < r_{3}

Reason:

△ A B C, r_{1} r_{2} + r_{2} r_{3} + r_{3} r_{1} = \frac{r_{1} r_{2} r_{3}}{r}

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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 traingle and r be the in-radius.Find the value of r1+r2+r3−r

The correct option is D 4RWe know, r1+r2+r3−r=Δs−a+Δs−b+Δs−c−Δs=Δ[s−b+s−a(s−a)(s−b)+(s−s+c)s(s−c)] =Δ[c(s−a)(s−b)+cs(s−c)]=Δc[s(s−c)+(s−a)(s−b)s(s−a)(s−b)(s−c)] =ΔcΔ2[2s2−s(a+b+c)+ab] =(cΔ)[2s2−s(2s)+ab]=4(abc4Δ)=4R

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 traingle and $r$ be the in-radius.
Find the value of $r_{1} + r_{2} + r_{3} - r$