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Section Formula

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

Δ

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2 Δ

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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 B $2 Δ$
$r \cdot r_{1} \cdot r_{2} \cdot r_{3} = \frac{△}{s} \times \frac{△}{s - a} \times \frac{△}{s - b} \times \frac{△}{s - c} \Rightarrow \frac{△^{4}}{s (s - a) (s - b) (s - c)}$
$[r = \frac{△}{s}, r_{1} = \frac{△}{s - a}, r_{2} = \frac{△}{s - b}, r_{3} = \frac{△}{s - c}]$
$r \cdot r_{1} \cdot r_{2} \cdot r_{3} = \frac{△^{4}}{△^{2}} = △^{2}$
Hence, $2 △$ is the correct answer.

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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

△ 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. Assertion :In a

Δ

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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. In a

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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 B 2Δr⋅r1⋅r2⋅r3=△s×△s−a×△s−b×△s−c⇒△4s(s−a)(s−b)(s−c)[r=△s,r1=△s−a,r2=△s−b,r3=△s−c]r⋅r1⋅r2⋅r3=△4△2=△2Hence, 2△ is the correct answer.

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