In a - ABC- the value of the product of r - r1 - r2 - r3 where r is the inradius and r1-r2-r3 are the exradius respectively is

The correct option is B Δ^2 ⇒ rr_1r_2r_3= ΔsΔs aΔs bΔs c #160; #160; #160; #160; #160; #160; #160; #160; #160; #160; ...

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Byju's Answer

Standard VII

Mathematics

Angle Sum Property

In a Δ ABC,...

Question

In a $Δ A B C$ , the value of the product of $(r \cdot r_{1} \cdot r_{2} \cdot r_{3})$ (where $r$ is the inradius and $r_{1}, r_{2}, r_{3}$ are the exradius respectively) is

R^{2}

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Δ^{2}

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s^{2}

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None

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Solution

The correct option is B $Δ^{2}$
$\Rightarrow r r_{1} r_{2} r_{3} = \frac{Δ}{s} \frac{Δ}{s - a} \frac{Δ}{s - b} \frac{Δ}{s - c}$
$= \frac{Δ^{4}}{s (s - a) (s - b) (s - c)}$
$= \frac{Δ^{4}}{Δ^{2}}$
$= Δ^{2}$ .

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Similar questions

Q. Prove that :

a (r r_{1} + r_{2} r_{3}) = b (r r_{2} + r_{3} r_{1}) = c (r r_{3} + r_{1} r_{2})

where

r

is inradius and

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

are exradius of triangle

A B C

and

a, b, c

are the corresponding sides.

Q. In a

Δ A B C,

the inradius and 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

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. 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?

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}

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In a ΔABC, the value of the product of (r⋅r1⋅r2⋅r3) (where r is the inradius and r1,r2,r3 are the exradius respectively) is

The correct option is B Δ2⇒rr1r2r3=ΔsΔs−aΔs−bΔs−c =Δ4s(s−a)(s−b)(s−c) =Δ4Δ2 =Δ2 .

In a $Δ A B C$ , the value of the product of $(r \cdot r_{1} \cdot r_{2} \cdot r_{3})$ (where $r$ is the inradius and $r_{1}, r_{2}, r_{3}$ are the exradius respectively) is

The correct option is B $Δ^{2}$
$\Rightarrow r r_{1} r_{2} r_{3} = \frac{Δ}{s} \frac{Δ}{s - a} \frac{Δ}{s - b} \frac{Δ}{s - c}$
$= \frac{Δ^{4}}{s (s - a) (s - b) (s - c)}$
$= \frac{Δ^{4}}{Δ^{2}}$
$= Δ^{2}$ .