Q1. The distance between two parallel tangents of a circle of radius 5 cm is:
Q2. A circle is inscribed in a triangle with sides 8 cm, 15 cm and 17 cm. The radius of the circle is:
Q3. If the angle between two tangents drawn from an external point P to a circle of radius $a$ and centre O is $60^\circ$, then $OP = $
Q4. Assertion (A): In a circle, the tangent is perpendicular to the radius at the point of contact. Reason (R): The length of tangents drawn from an external point to a circle are equal.
Q5. $XP$ and $XQ$ are tangents from an external point $X$ to a circle. Another tangent $AB$ touches the circle at $R$ and intersects $XP$ and $XQ$ at $A$ and $B$ respectively. Prove that $XA + AR = XB + BR$.
Solve on white paper.
Q6. Two tangents $TP$ and $TQ$ are drawn to a circle with centre $O$ from an external point $T$. Prove that $\angle PTQ = 2\angle OPQ$.
Q7. A circle is inscribed in a $\triangle ABC$ touching $AB, BC$ and $AC$ at $P, Q$ and $R$ respectively. If $AB = 10$ cm, $AR = 7$ cm and $CR = 5$ cm, find the length of $BC$.
Q8. $PQ$ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at $P$ and $Q$ intersect at a point $T$. Find the length $TP$.
Q9. Prove that the parallelogram circumscribing a circle is a rhombus.
Q10. A triangle $ABC$ is drawn to circumscribe a circle of radius 4 cm such that the segments $BD$ and $DC$ into which $BC$ is divided by the point of contact $D$ are of lengths 8 cm and 6 cm respectively. Find the sides $AB$ and $AC$.
Q11. Case Study:
A Ferris wheel has a radius of 10 m. The center of the wheel is 12 m above the ground. A safety beam is installed which acts as a tangent to the wheel from a point on the ground 24 m away from the point directly below the center of the wheel.
(i) What is the distance of the point on the ground from the center of the wheel?
(ii) Calculate the length of the safety beam (tangent) from the ground point to the point of contact.
(iii) If the angle of elevation of the center from the ground point is $\theta$, find $\tan \theta$.