Chapter 10: Circles

A Ferris wheel is a circular amusement ride. Suppose a Ferris wheel is represented by a circle with centre O. A person standing at a point P on the ground observes two points A and B on the Ferris wheel such that PA and PB are tangents to the circle.

1. If the angle between the two tangents is $60^\circ$, find the angle subtended by the chord AB at the centre.
   (A) $60^\circ$

   (B) $90^\circ$

   (C) $120^\circ$

   (D) $180^\circ$
   Show Solution
   Solution: (C) $120^\circ$
   Reason: The angle between tangents and the angle subtended by the chord at the centre are supplementary. $\angle AOB + \angle APB = 180^\circ \Rightarrow \angle AOB = 180^\circ - 60^\circ = 120^\circ$.
2. If the radius of the wheel is 10 m and the distance of P from the centre O is 20 m, find the length of the tangent PA.
   (A) 10 m

   (B) $10\sqrt{3}$ m

   (C) 20 m

   (D) $20\sqrt{3}$ m
   Show Solution
   Solution: (B) $10\sqrt{3}$ m
   Step 1: In right $\Delta OAP$, $OP^2 = OA^2 + PA^2$.
   Step 2: $20^2 = 10^2 + PA^2 \Rightarrow 400 = 100 + PA^2 \Rightarrow PA^2 = 300$.
   Step 3: $PA = \sqrt{300} = 10\sqrt{3}$ m.

Two concentric circles are of radii 5 cm and 3 cm. A chord of the larger circle touches the smaller circle.

1. Find the length of the chord of the larger circle which touches the smaller circle.
   (A) 4 cm

   (B) 5 cm

   (C) 8 cm

   (D) 10 cm
   Show Solution
   Solution: (C) 8 cm
   Step 1: Let the chord be AB and point of contact be P. OP $\perp$ AB.
   Step 2: In $\Delta OPA$, $OA=5, OP=3$. $AP = \sqrt{5^2 - 3^2} = \sqrt{16} = 4$ cm.
   Step 3: Length of chord $AB = 2 \times AP = 8$ cm.
2. What is the area of the ring formed between the two circles?
   (A) $16\pi$ cm$^2$

   (B) $25\pi$ cm$^2$

   (C) $9\pi$ cm$^2$

   (D) $34\pi$ cm$^2$
   Show Solution
   Solution: (A) $16\pi$ cm$^2$
   Step 1: Area = $\pi(R^2 - r^2) = \pi(5^2 - 3^2)$.
   Step 2: $\pi(25 - 9) = 16\pi$ cm$^2$.

A circle is inscribed in a quadrilateral ABCD touching its sides AB, BC, CD and DA at P, Q, R and S respectively.

1. If $AB = 6$ cm, $BC = 7$ cm and $CD = 4$ cm, find the length of $AD$.
   (A) 3 cm

   (B) 4 cm

   (C) 5 cm

   (D) 6 cm
   Show Solution
   Solution: (A) 3 cm
   Reason: For a circumscribed quadrilateral, $AB + CD = AD + BC$.
   $6 + 4 = AD + 7 \Rightarrow 10 = AD + 7 \Rightarrow AD = 3$ cm.
2. If the angles subtended by opposite sides at the centre are supplementary, and $\angle AOB = 100^\circ$, find $\angle COD$.
   (A) $80^\circ$

   (B) $100^\circ$

   (C) $90^\circ$

   (D) $120^\circ$
   Show Solution
   Solution: (A) $80^\circ$
   Reason: Opposite sides subtend supplementary angles at the centre. $\angle AOB + \angle COD = 180^\circ$.
   $100^\circ + \angle COD = 180^\circ \Rightarrow \angle COD = 80^\circ$.