Question : In the given figure $\mathrm{PQRS}$, a square of side $20\;\mathrm{cm}$ and $\mathrm{SR}$ is extended to point $\mathrm{T}$. If the length of $\mathrm{QT}$ is $25\;\mathrm{cm}$, then what is the distance (in $\mathrm{cm}$) between the centres, $\mathrm{O_1}$ and $\mathrm{O_2}$_, of the two circles?

Option 1: $5\sqrt{10}$

Option 2: $4\sqrt{10}$

Option 3: $2\sqrt{10}$

Option 4: $16\sqrt{2}$

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: $10\sqrt{2}\ \mathrm{~cm}$

Question : The distance between the centres of two circles having radii 22 cm and 18 cm, is 32 cm. The length (in cm) of the direct common tangent of the two circles is:

Option 1: $2 \sqrt{252} \mathrm{~cm}$

Option 2: $2 \sqrt{152} \mathrm{~cm}$

Option 3: $3 \sqrt{242} \mathrm{~cm}$

Option 4: $3 \sqrt{252} \mathrm{~cm}$

Question : Two circles of the same radius 6 cm, intersect each other at P and Q. If PQ = 10 cm, then what is the distance between the centres of the two circles?

Option 1: $10\mathrm{~cm}$

Option 2: $8\mathrm{~cm}$

Option 3: $6\sqrt{11} \mathrm{~cm}$

Option 4: $2\sqrt{11} \mathrm{~cm}$

Question : Two circles of radius 10 cm and 5 cm touch each other externally at point A. PQ is the direct common tangent of those two circles of centres O₁ and O₂, respectively. The length of PQ is equal to:

Option 1: $10\sqrt{2}$ cm

Option 2: $8\sqrt{2}$ cm

Option 3: $9\sqrt{2}$ cm

Option 4: $6\sqrt{2}$ cm

Question : Three circles of radius 6 cm each touch each other externally. Then the distance of the centre of one circle from the line joining the centres of the other two circles is equal to:

Option 1: $6\sqrt{5}\;\mathrm{cm}$

Option 2: $6\sqrt{3}\;\mathrm{cm}$

Option 3: $6\sqrt{2}\;\mathrm{cm}$

Option 4: $6\sqrt{7}\;\mathrm{cm}$