# CAT Practice : Coordinate Geometry

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Radius of the circle inscribed inside a semi - circle

## Inscribed Circles

Q.37: A circle is inscribed in a semi-circle as shown:-

The radius of the circle is:-
1. ${\frac{\surd {2}+ 1}{2}}$
2. ${\surd {2} - \frac{1}{2}}$
3. ${1 - \surd 2}$
4. ${\surd {2} - 1}$

Choice (D). ${\surd {2} - 1}$

## Detailed Solution

Area of semicircle =) $\frac{(π * 1^2)}{2}$ = 0.5 π

Area of circle should be less than the area of semicircle:-

Area = $π * (\frac{ (√2+1)}{2})^2 = \frac{π}{4} * (2+1+2√2) = (0.75+√2) * π > 0.5 π$

Area = $π * (√2 - \frac {1}{2})^2 = π * (2 + \frac{1}{4} - √2) = 0.85 π > 0.5 π$

Area = ${π * (1-2√2)^2 = π * (1+ 8-4√2) = (9-5.6) π = 3.4π > 0.5 π}$

Area = ${π * (√2-1)^2 = π * (2+ 1-2√2) = 0.2π}$

Correct Answer: (D). = ${\surd {2} - 1}$

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Geometry is probably the most vital topic as far as CAT preparation is concerned. Geometry sets the stage for Trigonometry, Cogeo and Mensuration as well.