If $A, A_{1}, A_{2}, A_{3}$ are the areas of the inscribed and escribed of a $△ A B C,$ then:

\sqrt{A_{1}} + \sqrt{A_{2}} + \sqrt{A_{3}} = \sqrt{π} (r_{1} + r_{2} + r_{2})

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\frac{1}{\sqrt{A_{1}}} + \frac{1}{\sqrt{A_{2}}} + \frac{1}{\sqrt{A_{3}}} = \frac{1}{\sqrt{A}}

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\frac{1}{\sqrt{A_{1}}} + \frac{1}{\sqrt{A_{2}}} + \frac{1}{\sqrt{A_{3}}} = \frac{s^{2}}{\sqrt{π} r_{1} r_{2} r_{3}}

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\sqrt{A_{1}} + \sqrt{A_{2}} + \sqrt{A_{3}} = \sqrt{π} (4 R + r)

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Solution

The correct options are
A $\sqrt{A_{1}} + \sqrt{A_{2}} + \sqrt{A_{3}} = \sqrt{π} (r_{1} + r_{2} + r_{2})$
B $\frac{1}{\sqrt{A_{1}}} + \frac{1}{\sqrt{A_{2}}} + \frac{1}{\sqrt{A_{3}}} = \frac{1}{\sqrt{A}}$
C $\frac{1}{\sqrt{A_{1}}} + \frac{1}{\sqrt{A_{2}}} + \frac{1}{\sqrt{A_{3}}} = \frac{s^{2}}{\sqrt{π} r_{1} r_{2} r_{3}}$
D $\sqrt{A_{1}} + \sqrt{A_{2}} + \sqrt{A_{3}} = \sqrt{π} (4 R + r)$
$∵ A = π r^{2}, A_{1} = π r_{1}^{2}, A_{2} = π r_{2}^{2}$ and $A_{3} = π r_{3}^{2}$
Option $(a)$
$\sqrt{A_{1}} + \sqrt{A_{2}} + \sqrt{A_{3}} = \sqrt{π} (r_{1} + r_{2} + r_{3})$
Option $(b)$
$\frac{1}{\sqrt{A_{1}}} + \frac{1}{\sqrt{A_{2}}} + \frac{1}{\sqrt{A_{3}}} = \frac{1}{\sqrt{π}} (\frac{1}{r_{1}} + \frac{1}{r_{2}} + \frac{1}{r_{3}})$
$= \frac{1}{\sqrt{π}} (\frac{1}{r})$
$= \frac{1}{\sqrt{π r^{2}}} = \frac{1}{\sqrt{A}}$
Option $(c)$
$∵ \frac{s^{2}}{\sqrt{π} r_{1} r_{2} r_{3}} = \frac{s^{2}}{\sqrt{π} \frac{△^{3}}{(s - a) (s - b) (s - c)}}$
$= \frac{s^{2} (s - a) (s - b) (s - c)}{\sqrt{π} △^{3}}$
$= \frac{s}{△ \sqrt{π}}$ where $△ = s (s - a) (s - b) (s - c)$
$= \frac{1}{r \sqrt{π}}$
$= \frac{1}{\sqrt{π r^{2}}}$
$= \frac{1}{\sqrt{A}}$
$= \frac{1}{\sqrt{A_{1}}} + \frac{1}{\sqrt{A_{2}}} + \frac{1}{\sqrt{A_{3}}}$
Option $(d)$
$∵ \sqrt{A_{1}} + \sqrt{A_{2}} + \sqrt{A_{3}} = \sqrt{π} (r_{1} + r_{2} + r_{3})$
$= \sqrt{π} (4 R + r)$

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Similar questions

\frac{a_{1}}{a_{1}} + \frac{a_{2}}{a_{2}} + \frac{a_{3}}{a_{3}} + \dots \frac{a_{n}}{a_{n}} = \frac{1}{1} + \frac{1}{a_{1}} + \frac{a_{1}}{a_{2}} + \frac{a_{2}}{a_{3}} + \dots \frac{a_{n - 2}}{a_{n - 1}}

A_{1}, A_{2},

A_{3}

R^{2}

A_{1} = {(x, y) : x \geq 0, y \geq 0, 2 x + 2 y - x^{2} - y^{2} > 1 > x + y}, A_{2} = {(x, y) : x \geq 0, y \geq 0, x + y > 1 > x^{2} + y^{2}}, A_{3} = {(x, y) : x \geq 0, y \geq 0, x + y > 1 > x^{3} + y^{3}} .

| A_{1} |, | A_{2} |,

| A_{3} |

A_{1}, A_{2}

A_{3}

a_{1}, a_{2}, a_{3}, . . . . a_{n}

\frac{a_{1}}{a_{2} + a_{3} + . . . . + a_{n}}, \frac{a_{2}}{a_{1} + a_{3} + . . . . + a_{n}}, . . . ., \frac{a_{n}}{a_{1} + a_{2} + . . . . + a_{n - 1}}

a_{1}, a_{2}, a_{3}, . . . . . a_{n}

\frac{a_{1}}{a_{2} + a_{3} + . . . a_{n}}, \frac{a_{2}}{a_{1} + a_{3} + . . . . + a_{n}}, \frac{a_{n}}{a_{1} + a_{2} + . . . + a_{n - 1}}

\frac{a_{1}}{a_{1} + 1 -} \frac{a_{2}}{a_{2} + 1 -} \frac{a_{3}}{a_{3} + 1 -} \dots, a_{1}, a_{2}, a_{3}, \dots

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