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Relation between Inradius and Perimeter of Triangle

In a Δ ABC,...

Question

In a $Δ A B C,$ the inradius and three exradii are $r, r_{1^{'}} r_{2}$ and $r_{3}$ respectively.
In usual notations the value of $r . r_{1} . r_{2} . r_{3}$ is equal to?

2 Δ

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Δ^{2}

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\frac{a b c}{4 R}

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none of these

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Solution

The correct option is C $Δ^{2}$
$r = \frac{Δ}{s}$

$r_{1} = \frac{Δ}{s - a}$

$r_{2} = \frac{Δ}{s - b}$

$r_{3} = \frac{Δ}{s - c}$

where, $Δ^{2} = s (s - a) (s - b) (s - c)$ (1)

Now,, $r . r_{1} . r_{2} . r_{3} = \frac{Δ}{s} . \frac{Δ}{s - a} . \frac{Δ}{s - b} . \frac{Δ}{s - c}$

$r . r_{1} . r_{2} . r_{3} = \frac{Δ^{4}}{Δ^{2}}$ (From 1)

$r . r_{1} . r_{2} . r_{3} = Δ^{2}$

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Similar questions

Q. In a

Δ A B C,

the inradius and three exradii are

r, r_{1,} r_{2}

and

r_{3}

respectively, in usual notations. The value of

r . r_{1} . r_{2} . r_{3}

is equal to

In a $△$ ABC, the inradius is r and three exradii are $r_{1}, r_{2} a n d r_{3}$ respectably. In usual notations the value of $r . r_{1} . r_{2} . r_{3}$ is equal to

Q. Assertion :In a

Δ

ABC,

r_{1} + r_{2} + r_{3} - r = 4 R

Reason:

r_{1} r_{2} + r_{2} r_{3} + r_{3} r_{1} = Δ^{2}

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}

Q. If in a

△ A B C, r_{1} r_{2} + r_{2} r_{3} + r_{3} r_{1}

is equal to
(where

r_{1}, r_{2}

and

r_{3}

are the exradii and

2 s

is the perimeter.)

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In a ΔABC, the inradius and three exradii are r,r1′r2 and r3 respectively.In usual notations the value of r.r1.r2.r3 is equal to?

The correct option is C Δ2r=Δs r1=Δs−a r2=Δs−b r3=Δs−c where, Δ2=s(s−a)(s−b)(s−c) (1) Now,, r.r1.r2.r3=Δs.Δs−a.Δs−b.Δs−c r.r1.r2.r3=Δ4Δ2 (From 1) r.r1.r2.r3=Δ2

In a $Δ A B C,$ the inradius and three exradii are $r, r_{1^{'}} r_{2}$ and $r_{3}$ respectively.
In usual notations the value of $r . r_{1} . r_{2} . r_{3}$ is equal to?