Question

If ABC is a quarter circle and circle is inscribed in it and if AB = 1, find the radius of smaller circle. Give proper explanation.​

Answer 1

Hello

Here , Radius of big Circle is 1 then we can say r is the radius of circle in it.

Diagonal of a square made

r = r*✓2

So,

R = 1 = r*✓2 + r

r = 1/(✓2+1) = ✓2 -1

radius of smaller circle is √2 - 1

Answer 2

$\Large\underline{\underline{\sf{Given}:}}$

• ABC is a Quarter circle ...
• AB = 1 unit .

$\Large\underline\mathfrak{Question}$

• Find radius of smaller circle ..

__________________

$\Large\underline{\underline{\sf{Solution}:}}$

$\orange{\textbf{ReFer to image First}}$

• since ABC is a quadrant of a circle ,, so, AB = BC = BP = radius of Quadrant = 1 unit ..
• ER = RD = BD = EB = radius of smaller circle = r units.
• Hence, ERBD is a square .

we know that,

$\pink{ \textbf{Diagonal of square} = \large\boxed{\bold{ \sqrt{2} \times side }}}$

Hence ,

→ BR = (√2×r) units ..

Now, we know that,

BP = BR + RP = radius of Quadrant = 1 unit ..

Putting values we get,,,

$\red\leadsto \: \blue{1 = (\sqrt{2} \: r \: + r)} \\ \\ \red\leadsto \: \green{1 = r( \sqrt{2} + 1)} \\ \\ \red\leadsto \: r = \frac{1}{ \sqrt{2} + 1} \\ \\ \textbf{Rationalize the Denominator} \\ \\ \red\leadsto \: r = \orange{ \frac{ \sqrt{2} - 1 }{2 - 1}} \\ \\ \red\leadsto \: \large\boxed{\bold{r = \sqrt{2} - 1 }}$

Hence, radius of smaller circle (r) is equal to (√2 - 1) units .

$\red{\large\underline\textbf{Hope it Helps You.}}$