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Area of a Triangle

In Δ ABC ...

Question

In $Δ$ $A B C$ 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?

\frac{5}{4}

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\frac{4}{5}

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\frac{7}{4}

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\frac{4}{7}

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Solution

The correct option is A $\frac{5}{4}$
$r_{1} = 2 r_{2} = 3 r_{3}$
$\Rightarrow \frac{△}{s - a} = \frac{2 △}{s - b} = \frac{3 △}{s - c}$
$\Rightarrow s - b = 2 (s - a)$
$\Rightarrow \frac{a + b + c}{2} - b = 2 (\frac{a + b + c}{2} - a)$
$\Rightarrow 3 a - 3 b = c$ ...(1)
$\Rightarrow s - c = 3 (s - a)$
$\Rightarrow \frac{a + b + c}{2} - c = 3 (\frac{a + b + c}{2} - a)$
$\Rightarrow a - \frac{b}{2} = c$ ...(2)
from (1) and (2): $3 a - 3 b = a - \frac{b}{2}$
$\Rightarrow \frac{a}{b} = \frac{5}{4}$
Ans: A

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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.

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

, then the perimeter of the triangle 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.

The value of

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

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

then

a : b

is equal to?

Δ

A B C

the sides opposite to angles

A, B, C

are denoted by

a, b, c

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r_{3}

be the ex-radii of the traingle and

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Δ

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the sides opposite to angles

A, B, C

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a, b, c

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r

=

in-radius, R

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are ex-radii)

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In Δ ABC the sides opposite to angles A,B,C are denoted by a,b,c respectively. r1,r2,r3 are the ex-radii then if r1=2r2=3r3, the value of a:b is equal to?

The correct option is A 54r1=2r2=3r3⇒△s−a=2△s−b=3△s−c⇒s−b=2(s−a)⇒a+b+c2−b=2(a+b+c2−a)⇒3a−3b=c ...(1)⇒s−c=3(s−a)⇒a+b+c2−c=3(a+b+c2−a)⇒a−b2=c ...(2)from (1) and (2): 3a−3b=a−b2⇒ab=54Ans: A

In $Δ$ $A B C$ 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?