1) In the figure 1, two tangents TP and TQ are drawn to a circle with centre O from an external point
T, prove that ∠PTQ = 2OPQ.
2) In the figure 2 , from an external point P, two tangents PT and PS are
drawn to a circle with centre O and radius r. If OP = 2r, show that
∠OTS =∠OST = 30°. Fig- 1
3) In the given figure 3, a circle touches the side DF of AEDF at H and
touches ED and EF produced at K&M respectively. If EK = 9 cm,
calculate the perimeter of AEDF (in cm).
4) In the given figure 4, PQ is a chord of a circle with centre O and PT is a
Fig- 2
tangent. If ∠QPT = 60°, find ∠PRQ.
2122d
sd
5) A circle is inscribed in a quadrilateral ABCD touching its sides AB, BC,
CD and AD at P, Q, R and S respectively. If the radius DA of the circle is 10
cm, BC = 38 cm, PB = 27 cm and AD ⊥ CD, then calculate the length of CD
6) In the figure 6, AB is the diameter of a circle with centre O and AT is a
tangent. If ∠AOQ = 58°, find ∠ATQ.
Fig- 3
7) Two concentric circles are of radii 7 cm and r cm respectively, where r > 7. A
chord of the larger circle, of length 48 cm, touches the smaller circle. Find the
value of r.
Fig- 4
8) Prove that the parallelogram circumscribing a circle is a rhombus.
9) In figure 5, a quadrilateral ABCD is drawn to circum- DA scribe a circle, with
centre O, in such a way that the sides AB, BC, CD and DA touch the circle at
the points P, Q, RA and S respectively. Prove that: AB + CD = BC + DA
10)In Figure 6, a right triangle ABC, circumscribes a circle of radius r. If AB and BC
are of lengths 8 cm and 6 cm respectively, find the value of r.
11) If from an external point P of a circle with centre O, two tangents PQ and PR Fig- 5
are drawn such that ∠QPR = 120°, prove that 2PQ = PO
12) From a point T outside a circle of centre O, tangents TP and TQ are drawn to
the circle. Prove that OT is the right bisector of line segment P
13) Prove that the tangent at any point of a circle is perpendicular to the radius
through the point of contact.
Fig- 7
14)In the given figure 7 , a circle inscribed in ∆ABC touches its sides AB, BC and AC at points D, E & F
K respectively. If AB = 12 cm, BC = 8 cm and AC = 10 cm, then find the lengths of AD, BE and CF.
15) In figure 8, PQ is a chord of length 16 cm, of a circle of radius 10 cm.
The tangents at P and Q intersect at a point T. Find the length of TP
16)In the figure 9, tangents PQ and PR are drawn from an external point P
to a circle with centre O, such that ∠RPQ = 30°. A chord RS is drawn
parallel to the tangent PQuestion Find ∠RQS.
17) Prove that the tangent drawn at the mid-point of an arc of a circle is
Fig- 7
parallel to the chord joining the end points of the arc
18)In the figure 10 , two equal circles, with centres 0 and O’, touch each
other A at X. OO’ produced meets the circle with centre O’ at A. AC is
Fig-8
tangent to the circle with centre O, at the point C. O’D is perpendicular
to AC. Find the value of DO′CO.
19) In the figure 11 , the sides AB, BC and CA of triangle ABC touch a circle with
centre o and radius r at P, Q and R. respectively.
Prove that: Fig-9
(i) AB + CQ = AC + BQ
(ii) Area (AABC) = 12 (Perimeter of ∆ABC ) × r
Fig- 10
Fig - 11
