find the radius of a circle which is inscribed in a quadrent of a circle of radius 2under root 2 - Maths - Areas Related to Circles

Question

find the radius of a circle which is inscribed in a quadrent of
a circle of radius 2under root 2 and touching to both the radius of
a circle as well touching the length of arc of a quadrent

Lalit Mehra · Accepted Answer

Given, Radius of quadrant of the circle = $2 \sqrt{2}$

Let the radius of the inscribed circle be r.

Let O and O₁ be the centre of the quadrant of the circle and inscribed circle respectively.

∠O₁AO = ∠O₁BO = 90° (Radius is perpendicular to the tangent at point of contact)

∴ OAO₁B is a square.

⇒ O₁A = OA = r

In right ΔOAO₁,

OO₁² = OA² + O₁A²

∴ OO₁² = r² + r² = 2r²

$\Rightarrow {OO}_{1} = \sqrt{2} r$

When two circles touches each other internally, then the distance between their centres is equal to difference of their radius.

$∴ {OO}_{1} = 2 \sqrt{2} - r \Rightarrow \sqrt{2} r = 2 \sqrt{2} - r \Rightarrow \sqrt{2} r + r = 2 \sqrt{2} \Rightarrow (\sqrt{2} + 1) r = 2 \sqrt{2} \Rightarrow r = \frac{2 \sqrt{2}}{\sqrt{2} + 1}$

Thus, the radius of the inscribed circle is $\frac{2 \sqrt{2}}{\sqrt{2} + 1}$ units.

find the radius of a circle which is inscribed in a quadrent of a circle of radius 2under root 2 and touching to both the radius of a circle as well touching the length of arc of a quadrent.