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Definition of Circle

In aright ang...

Question

In aright angles $△ A B C$ of maximum area is inscribed within a cricle pf radius $R$ then

Δ = 2 R^{2}

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\frac{1}{r_{1}} + \frac{1}{r_{2}} + \frac{1}{r_{3}} = \frac{(\sqrt{2} + 1)}{R}

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r = (\sqrt{2} - 1) R

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s = (1 + \sqrt{2}) R

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Solution

The correct options are
B $s = (1 + \sqrt{2}) R$
D $\frac{1}{r_{1}} + \frac{1}{r_{2}} + \frac{1}{r_{3}} = \frac{(\sqrt{2} + 1)}{R}$
For a right-angled inscribed in a circle of radius $R$ , the length of the hypotenuse is $2 R$

$∴$ the area is maximum when the triangle is isosceles with each side $= \sqrt{2} R$

$∴ s = \frac{1}{2} (2 \sqrt{2} + 2) R = (\sqrt{2} + 1) R$

$∴ Δ = \frac{1}{2} \sqrt{2} R . \sqrt{2} R = R^{2} \Rightarrow \frac{1}{r} = \frac{(\sqrt{2} + 1)}{R}$

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

$= \frac{s}{Δ} = \frac{1}{r} = \frac{\sqrt{2} + 1}{R}$

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Similar questions

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