Question : In the given figure, $PQRS$ is a rectangle and a semicircle with $SR$ as the diameter is drawn. A circle is drawn as shown in the figure. If $QR=7\;\mathrm{cm}$, then what is the radius (in $\mathrm{cm}$) of the small circle?
Option 1: $21+14\sqrt{2}$
Option 2: $21-14\sqrt{2}$
Option 3: both $21+14\sqrt{2}$ and $21-14\sqrt{2}$
Option 4: None of these
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Correct Answer: $21-14\sqrt{2}$
Solution :
We have $PQRS$ as a rectangle and a semicircle with $SR$ as diameter and $QR = 7\;\mathrm{cm}$. Construction: Draw a line from $Q$ on line $ST$ (where $T$ is the centre of the semicircle) and pass through point $U$ (where $U$ is the point of contact and $O$ is the centre of the circle). Now, $RT$ and $QR$ are the radius of the bigger circle and equal to $7\;\mathrm{cm}$. In $\triangle QRT$, $QT^2=QR^2+TR^2$ $QT^2 = 49 + 49$ $ QT = 7\sqrt{2}$ ____(i) Let the radius of the smaller circle $=r\;\mathrm{cm}$ $QO = r\sqrt{2}$ From the figure, $QT = TU + UO + OQ$ $QT = 7 + r + r\sqrt{2}$ ____(ii) From equation (i) and (ii) $7\sqrt{2} = 7 + r (\sqrt{2} + 1)$ ⇒ $r = \frac{7 (\sqrt{2} - 1)}{(\sqrt{2} + 1)}$ ⇒ $r = 7 (3 - 2\sqrt{2})$ Hence, the correct answer is $21 - 14\sqrt{2}$.
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Question : In a circle, a diameter AB and a chord PQ (which is not a diameter) intersect each other at X perpendicularly. If AX : BX = 3 : 2 and the radius of the circle is 5 cm, then the length of the chord PQ is:
Option 1: $2\sqrt{13}\;\mathrm{cm}$
Option 2: $5\sqrt{3}\;\mathrm{cm}$
Option 3: $4\sqrt{6}\;\mathrm{cm}$
Option 4: $6\sqrt{5}\;\mathrm{cm}$
Question : The area of a sector of a circle is 88 sq. cm., and the angle of the sector is 45°. Find the radius of the circle. (Use $\pi=\frac{22}{7}$)
Option 1: $3 \sqrt{ 11} \mathrm{~cm}$
Option 2: $4 \sqrt{ 14} \mathrm{~cm}$
Option 3: $6 \sqrt{ 13} \mathrm{~cm}$
Option 4: $5 \sqrt{ 14} \mathrm{~cm}$
Question : The area of a sector of a circle is 66 cm2 and the angle of the sector is 60°. Find the radius of the circle.
Option 1: $5 \sqrt{15} \mathrm{~cm}$
Option 2: $6 \sqrt{14} \mathrm{~cm}$
Option 3: $7 \sqrt{19} \mathrm{~cm} $
Option 4: $3 \sqrt{14} \mathrm{~cm} $
Question : If the area of a circle is $616 \mathrm{~cm}^2$ and a chord $XY=10 \mathrm{~cm}$, then find the perpendicular distance from the centre of the circle to the chord $XY$.
Option 1: $\sqrt{171} \mathrm{~cm}$
Option 2: $\sqrt{161 } \mathrm{~cm}$
Option 3: $\sqrt{117 } \mathrm{~cm}$
Option 4: $\sqrt{ 181} \mathrm{~cm}$
Question : What is the perimeter of a square inscribed in a circle of radius 5 cm?
Option 1: $20 \sqrt{2}\ \mathrm{~cm}$
Option 2: $40\sqrt{2}\ \mathrm{~cm}$
Option 3: $30\sqrt{2}\ \mathrm{~cm}$
Option 4: $10\sqrt{2}\ \mathrm{~cm}$
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