This document contains 15 geometry problems involving circles and tangents. Many of the problems relate to properties of circles such as angles inscribed by chords and tangents, relationships between radii and chords, properties of tangents drawn from external points, and circles inscribed or circumscribed in triangles and quadrilaterals. The problems are from past exams of the CBSE board in India and range from proving geometric theorems to calculating specific lengths or angles based on given information.

1. (1)From a point p, 10cm away from the centre of a
circle, a tangent PT of length 8cm is drawn. Find the
radius of the circle (CBSE 2002)
(2)In the given fig,AB is a chord of length 9.6cm of a
circle with centre O and radius 6cm.The tangent at A
and B intersect at P.Find the length of PA (CBSE 2009C)
(3) Two tangents PA and PB are drawn to a circle with
centre O from an external point P.Prove that
<APB=2<OAB (CBSE 2009)
(4) O is the centre of two concentric circles .AB is a
chord of the larger circle touching the smaller circle at
C.Prove that AC=BC
(CBSE 2009)
(5) Prove that the angle between the two tangents
drawn from an external point to a circle is
supplementary to the angle subtended by the line

2. segments joining the points of contact to the centre.
(CBSE 2008C)
(6) A circle is inscribed in a triangle ABC, touching BC,
CA and AB at P, Q and R respectively, as shown in the
given figure. If AB=10cm, AQ=7cm and CQ=5cm, then
find the length of BC
(CBSE 2009C)
(7) A circle is touching the side BC of triangle ABC at P
and touching AB and AC produced at Q and R
respectively. Prove that AQ=1/2(perimeter of triangle
ABC)
(CBSE 2001, 02, 06)
(8)In the given fig, ABC is a right –angled triangle with
AB=6cm and AC=8cm.A circle with centre O has been
inscribed inside the triangle. Calculate the value of
r,the radius of the inscribed circle. (CBSE 2006C)

3. (9) A quadrilateral ABCD is drawn to circumscribe a
circle, as shown in the figure. Prove that AB+CD=AD+BC
(CBSE 2008,’08C,’09)
(10) Prove that the parallelogram circumscribing a
circle is a rhombus. (CBSE 2008,’09,’10)
(11)In the given figure, a circle is inscribed in a
quadrilateral ABCD in which <B=90 .If AD =23cm,
AB=29cm and DS=5cm, find the radius of the circle
(CBSE 2008)
(12)In the given figure, PA is a tangent from an external
point P to a circle with centre O.If<POB=115, find
<APO. (CBSE 2009C)

4. (13) From a point P, two tangents PA and PB are drawn
to a circle C (O, r).If OP=2r, show that triangle APB is
equilateral (CBSE 2008)
(14) From an external point P,tangent PA and PB are
drawn to a circle with centre O.If CD is the tangent to
the circle at a point E and PA=14cm,find the perimeter
of
PCD
(CBSE 2002)
(15) A circle is inscribed in a triangle ABC touching AB,
BC and AC are P, Q and R respectively. If AB=10cm,
AR=7cm and CR=5cm, find the length of BC (CBSE
2009C)