Page 1 1. In the adjoining figure, a circle touches all the four sides of a quadrilateral ABCD whose sides are AB=6cm, BC=9cm and CD=8 cm. Find the length of side AD. Page 2 1. In the adjoining figure, a circle touches all the four sides of a quadrilateral ABCD whose sides are AB=6cm, BC=9cm and CD=8 cm. Find the length of side AD. Sol: We know that when a quadrilateral circumscribes a circle then sum of opposites sides is equal to the sum of other opposite sides. 6 8 9 5 AB CD AD BC AD AD cm 2. In the given figure, PA and PB are two tangents to the circle with centre O. If 50 APB then what is the measure of . OAB Sol: Construction: Join OB We know that the radius and tangent are perpendicular at their point of contact 90 OBP OAP Now, In quadrilateral AOBP 360 AOB OBP APB OAP [Angle sum property of a quadrilateral] 90 50 90 360 230 360 130 AOB BOC AOB Now, In isosceles triangle AOB 180 AOB OAB OBA [Angle sum property of a triangle] 130 2 180 25 OAB OAB OBA OAB 3. In the given figure, O is the centre of a circle. PT and PQ are tangents to the circle from an external point P. If 70 TPQ , find the . TRQ Page 3 1. In the adjoining figure, a circle touches all the four sides of a quadrilateral ABCD whose sides are AB=6cm, BC=9cm and CD=8 cm. Find the length of side AD. Sol: We know that when a quadrilateral circumscribes a circle then sum of opposites sides is equal to the sum of other opposite sides. 6 8 9 5 AB CD AD BC AD AD cm 2. In the given figure, PA and PB are two tangents to the circle with centre O. If 50 APB then what is the measure of . OAB Sol: Construction: Join OB We know that the radius and tangent are perpendicular at their point of contact 90 OBP OAP Now, In quadrilateral AOBP 360 AOB OBP APB OAP [Angle sum property of a quadrilateral] 90 50 90 360 230 360 130 AOB BOC AOB Now, In isosceles triangle AOB 180 AOB OAB OBA [Angle sum property of a triangle] 130 2 180 25 OAB OAB OBA OAB 3. In the given figure, O is the centre of a circle. PT and PQ are tangents to the circle from an external point P. If 70 TPQ , find the . TRQ Sol: Construction: Join OQ and OT We know that the radius and tangent are perpendicular at their point of contact 90 OTP OQP Now, In quadrilateral OQPT 360 QOT OTP OQP TPO [Angle sum property of a quadrilateral] 90 90 70 360 250 360 110 QOT QOT QOT We know that the angle subtended by an arc at the center is double the angle subtended by the arc at any point on the remaining part of the circle. 1 55 2 TRQ QOT 4. In the given figure common tangents AB and CD to the two circles with centres 1 2 O and O intersect at E. Prove that AB=CD. Sol: We know that tangent segments to a circle from the same external point are congruent. So, we have EA = EC for the circle having center 1 O and ED = EB for the circle having center 1 O Now, Adding ED on both sides in EA = EC. we get EA ED EC ED EA EB EC ED AB CD Page 4 1. In the adjoining figure, a circle touches all the four sides of a quadrilateral ABCD whose sides are AB=6cm, BC=9cm and CD=8 cm. Find the length of side AD. Sol: We know that when a quadrilateral circumscribes a circle then sum of opposites sides is equal to the sum of other opposite sides. 6 8 9 5 AB CD AD BC AD AD cm 2. In the given figure, PA and PB are two tangents to the circle with centre O. If 50 APB then what is the measure of . OAB Sol: Construction: Join OB We know that the radius and tangent are perpendicular at their point of contact 90 OBP OAP Now, In quadrilateral AOBP 360 AOB OBP APB OAP [Angle sum property of a quadrilateral] 90 50 90 360 230 360 130 AOB BOC AOB Now, In isosceles triangle AOB 180 AOB OAB OBA [Angle sum property of a triangle] 130 2 180 25 OAB OAB OBA OAB 3. In the given figure, O is the centre of a circle. PT and PQ are tangents to the circle from an external point P. If 70 TPQ , find the . TRQ Sol: Construction: Join OQ and OT We know that the radius and tangent are perpendicular at their point of contact 90 OTP OQP Now, In quadrilateral OQPT 360 QOT OTP OQP TPO [Angle sum property of a quadrilateral] 90 90 70 360 250 360 110 QOT QOT QOT We know that the angle subtended by an arc at the center is double the angle subtended by the arc at any point on the remaining part of the circle. 1 55 2 TRQ QOT 4. In the given figure common tangents AB and CD to the two circles with centres 1 2 O and O intersect at E. Prove that AB=CD. Sol: We know that tangent segments to a circle from the same external point are congruent. So, we have EA = EC for the circle having center 1 O and ED = EB for the circle having center 1 O Now, Adding ED on both sides in EA = EC. we get EA ED EC ED EA EB EC ED AB CD 5. If PT is a tangent to a circle with center O and PQ is a chord of the circle such that 70 QPT , then find the measure of . POQ Sol: We know that the radius and tangent are perpendicular at their point of contact. 90 OPT Now, 90 70 20 OPQ OPT TPQ Since, OP = OQ as both are radius 20 OPQ OQP (Angles opposite to equal sides are equal) Now, In isosceles POQ 180 POQ OPQ OQP (Angle sum property of a triangle) 180 20 140 POQ 6. In the given figure, a triangle ABC is drawn to circumscribe a circle of radius 2 cm such that the segments BD and DC into which BC is divided by the point of contact D, are of lengths 4cm and 3cm respectively. If the area of 2 21 ABC cm then find the lengths of sides AB and AC. Sol: Construction: Join OA, OB, OC, OE AB at E and OF AC at F We know that tangent segments to a circle from the same external point are congruent Now, we have , 4 AE AF BD BE cm and 3 CD CF cm Now, Area ABC Area BOC Area AOB Area AOC Page 5 1. In the adjoining figure, a circle touches all the four sides of a quadrilateral ABCD whose sides are AB=6cm, BC=9cm and CD=8 cm. Find the length of side AD. Sol: We know that when a quadrilateral circumscribes a circle then sum of opposites sides is equal to the sum of other opposite sides. 6 8 9 5 AB CD AD BC AD AD cm 2. In the given figure, PA and PB are two tangents to the circle with centre O. If 50 APB then what is the measure of . OAB Sol: Construction: Join OB We know that the radius and tangent are perpendicular at their point of contact 90 OBP OAP Now, In quadrilateral AOBP 360 AOB OBP APB OAP [Angle sum property of a quadrilateral] 90 50 90 360 230 360 130 AOB BOC AOB Now, In isosceles triangle AOB 180 AOB OAB OBA [Angle sum property of a triangle] 130 2 180 25 OAB OAB OBA OAB 3. In the given figure, O is the centre of a circle. PT and PQ are tangents to the circle from an external point P. If 70 TPQ , find the . TRQ Sol: Construction: Join OQ and OT We know that the radius and tangent are perpendicular at their point of contact 90 OTP OQP Now, In quadrilateral OQPT 360 QOT OTP OQP TPO [Angle sum property of a quadrilateral] 90 90 70 360 250 360 110 QOT QOT QOT We know that the angle subtended by an arc at the center is double the angle subtended by the arc at any point on the remaining part of the circle. 1 55 2 TRQ QOT 4. In the given figure common tangents AB and CD to the two circles with centres 1 2 O and O intersect at E. Prove that AB=CD. Sol: We know that tangent segments to a circle from the same external point are congruent. So, we have EA = EC for the circle having center 1 O and ED = EB for the circle having center 1 O Now, Adding ED on both sides in EA = EC. we get EA ED EC ED EA EB EC ED AB CD 5. If PT is a tangent to a circle with center O and PQ is a chord of the circle such that 70 QPT , then find the measure of . POQ Sol: We know that the radius and tangent are perpendicular at their point of contact. 90 OPT Now, 90 70 20 OPQ OPT TPQ Since, OP = OQ as both are radius 20 OPQ OQP (Angles opposite to equal sides are equal) Now, In isosceles POQ 180 POQ OPQ OQP (Angle sum property of a triangle) 180 20 140 POQ 6. In the given figure, a triangle ABC is drawn to circumscribe a circle of radius 2 cm such that the segments BD and DC into which BC is divided by the point of contact D, are of lengths 4cm and 3cm respectively. If the area of 2 21 ABC cm then find the lengths of sides AB and AC. Sol: Construction: Join OA, OB, OC, OE AB at E and OF AC at F We know that tangent segments to a circle from the same external point are congruent Now, we have , 4 AE AF BD BE cm and 3 CD CF cm Now, Area ABC Area BOC Area AOB Area AOC 1 1 1 21 2 2 2 42 7 2 4 2 3 2 21 7 4 3 BC OD AB OE AC OF x x x x 21 14 2 2 7 3.5 4 3.5 7.5 3 3.5 6.5 x x x cm AB cm and AC cm 7. Two concentric circles are of radii 5cm and 3cm. Find the length of the chord of the larger circle (in cm) which touches the smaller circle. Sol: Given Two circles have the same center O and AB is a chord of the larger circle touching the smaller circle at C; also. 5 OA cm and 3 OC cm In 2 2 2 , OAC OA OC AC 2 2 2 2 2 2 2 2 5 3 25 9 16 4 AC OA OC AC AC AC AC cm 2 AB AC (Since perpendicular drawn from the center of the circle bisects the chord) 2 4 8 AB cm The length of the chord of the larger circle is 8 cm. 8. Prove that the perpendicular at the point of contact of the tangent to a circle passes through the centre. Sol:Read More

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