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2020-21 Academic year Works, Mathematics X, Circles, , 16, ◦, , Concepts, , a) Angle in the semicircle is 90, , b) Angle outside the semicircle is less than 90◦, c) Angle inside the semicircle is greater than 90◦, Worksheet16, 1) ABCis a triangle in which AB = AC = BC, , a) What are the measure of its angles?, b) What is the position of the vertex C based on the circle with diametreAB?Is it inside ,outside or on, the circle., ◦, , Answers, , a) 60, , b) Outside the circle, 2) In the figure AB is the diametre of the circle.AC and P Dare perpendicular to CD, , 1, , Prepared by Sri John PA

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a) What is the measure of angle AP B?, b) Suggest a suitable name to ACDP, c) If AP = 8cm and BP = 6cm then what is the radius of this circle?, Answers, a) ∠AP B = 90 :Angle in the semicircle, ◦, , b) ∠AP D = 180 − 90 = 90◦ , Angles ofACDP are 90◦ ACDP is a square., √, c) AP B is a right trangle. AB = 82 + 62 = 10. Radius = 5cm, 3) O is the centre of the circle with diametre AB.Another circle is drawn with AOas the diametre, , a) What are the measure of ∠AP O, ∠ACB, b) Outer circle has radius 5cm and BC = 8cm . What is the length OP ?, c) Is AP = P C? Why?, d) What is the length of AC?, Answers, a) 90 . Reason ∠AP O, ∠ACB are the angles in the semicircle, ◦, , AO, =, b) Triangle AP O,triangle ACB are similar. AB, 5, OP, 10 = 8 , OP = 4cm, , OP, BP, , c) ACis the chord of big circle. OP is perpendicular from centre to this chord. OP bisect, AC.Therefore AP = P C, √, d) AP = 52 − 42 = 3cm.AC = 6cm, 4) Draw a circle of radius 3cm and construct a rectangle with vertices on the circle. One side of the rectangle, should be 4cm . What is the lengh of other side?(Write the measurement), Answers, a) Draw circle. Diametre AB = 6cm .Draw an of radius 4cm , centre at A which cut the circle, at C. Join AC and BC, b) Draw an of radius 4cm , centre at B which cut the circle at D in the other side of the diametre, at D. Join BD and AD, c) AngleACB, angleADB are 90◦ each. ACBDis a rectangle, √, √, √, d) Measure the other side. BC = 62 42 = 20 = 2 5cm, 5) Sides of triangle ABCare AB = 5cm, AC = 12cm ,BC = 13cm, , 2, , Prepared by Sri John PA

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a) What kind of triangle is this ?, b) What is the position of A based on the circle with diametre BC?, c) What is the position of C based on the circle with diametre AB?, d) What is the position of B based on the circle with diametre AB?, Answers, 2, , 2, , 2, , a) 5 + 12 = 13, This is a right triangle, b) In triangleABC angleA = 90◦ .A is on the circle, c) One angle is 90◦ and other two angles are less than 90◦ . ∠C < 90◦ ,therefore C is, outside the circle, d) ∠B < 90◦ .B is outside the circle., , [email protected], , 3, , Prepared by Sri John PA

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2020-21 Academic year Works, Mathematics X, Circles, , 17, Concepts, , ◦, , a) Angle in the semicircle is 90, , b) Angle outside the semicircle is less than 90◦, c) Angle inside the semicircle is greater than 90◦, Worksheet17, 1) In the figure AB is the diametre of a semicircle.Three angles x, y, z are marked outside, on the semicircle, and inside the semicircle., , a) What is the value of y?, b) If x, y, z are in an arithmetic sequence,then what is x + z?, c) If the common difference of the sequence is 50 then find x and z, a) y = 90, , Answers, , ◦, , b) x + z = 2 × 90 = 180◦ (Refer the propery of arithmetic sequence), c) d = 50 ∴ x = 90 − 50 = 40◦ , z = 90 − 50 = 140◦, 2), , a) Draw a circle of radius 3cm. Construct a square with vertices are on the circle., b) What is the length of its side?, c) Calculate the area of the square., Answers, a) Draw a circle of radius 3cm and diametreAB., b) Draw another diametre CDperpendicular to AB. Join the end points of the diametre ., ACBD will be a square, 1 √, √, c) Side of the√square√AC = 32 + 32 = 3 2cm., Area = 3 2 × 3 2 = 18sq.cm, , Prepared by Sri John PA

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3) In triangle ABC, AB = AC.A circle is drawn with one of these sides as diametre.Prove that the circle, biscts the side BC, Answers, a) Circle intersect BC at P . ∠BP A = 90◦ ., b) Triangle AP B and triangle AP C are equal triangles.BP = CP, 4) The sides of a triangle are, , √ √, √, 2, 3 and 5., , a) What kind of triangle is this?, b) What is the position of the vertex opposite to the side, diametre?, , √, 5based on a circle with this side as the, , c) What is the position of other two vertices based on this circle?, Answers, √ 2 √ 2, a) 2 + 3 = 5 . This is a right triagle., √, b) Angle opposite to the side of length 5 cm is 90◦ . The vertex opposite to this side is on, the circle., √, , 2, , c) Other two angles are less than 90◦ . Vertex is outside the circle, 5) ABC is an equilateral triangle.A semicircle is drawn with diametre AB. Semicircle intersect the sides at P, and Q., , a) What is the measure of angle BQC?(Draw angle in the figure), b) What are the angles of triangle ABQ and triangle BQC, c) Prove that the semicircle bisects the side AB and AC, Answers, a) Draw a rough figure.∠BQC = 90◦ .(Angle in the semicircle), b) ∠A = 60◦ , ∠ABQ = 30◦ , ∠AQB = 90◦ , ∠QBC = 30◦ , ∠C = 60◦, c) BQ is the altitude of the equilateral trangle. Therefore AQ = CQ, , [email protected], , 2, , Prepared by Sri John PA

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2020-21 Academic year Works, Mathematics X, Circles, , 18, Concepts, a) An arc of a circle can make three type of angles. Angle on the arc, angle at the centre and angle in, the complement, b) Angle formed by the arc in the complement is half the angle at the centre, c) Sum of the angles at the centre and in the complement is 180◦, d) Angles on an arc are equal, , Worksheet18, 1) In the figure ∠BDE = 40◦, , a) What is the measure of angleADB?, b) Wha is the measure of angleACB?, c) What is the measure of angle AOB?, , a) ∠ADB = 180 − 40 = 140, , ◦, , Answers, , b) ∠ACB = 180 − 140 = 40◦, c) ∠AOB = 2 × 40 = 80◦, , 1, , Prepared by Sri John PA

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2) Triangle OAB is an equilateral triangle, , a) What is the measure of angleAOB?, b) What is the measure of angleAP B?, c) What is the measure of angleAQB?, , a) ∠AOB = 60, b) ∠AP B =, , 1, 2, , Answers, , ◦, , × 60◦ = 30◦, , c) ∠AQB = 180 − 30 = 150◦, 3) Draw a circle of radus 3cm.Construct the angles 30◦ and 150◦ with vertices on the circle using compasses, and scale only., Answers, ⋆ Draw a circle of radius 3cm.Mark the center of the circle as O, ⋆ Mark a point A on the circle.Draw the radius OA., ⋆ With A as the centre and OA as radius , draw an arc which cut the circle at B. Join OB,, ∠AOB = 60◦, ⋆ Mark a point P on the complement of the arc AB,which makes 60◦ at the center.∠AP B =, 1, ◦, 2 × 60 = 30, ⋆ Mark a point Q on the arc AB. ∠AQB = 180 − 30 = 150◦, 4) In the figure ∠ABC, ∠AOC, ∠ADCare in an arithmetic sequence, , a) What is the relation between angle ABC and angle AOC, b) What is the relation between angle ABC and ADC, c) Find the measure of these angles, , 2, , Prepared by Sri John PA

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Answers, , a) ∠AOC = 2 × ∠ABC, b) ∠ABC + ∠ADC = 180◦, , c) Let ∠ABC = x, ∠AOC = y, ∠ADC = z, x, y, z are in an arithmetic sequence.Therefore 2y = x + z, From the relations noted above , y = 2x, x + z = 180, 2y = x + z, 2 × 2x = x + z = 180, 4x = 180, x = 45, , x = 45, y = 90, z = 135 ., ∠ABC = 45◦ , ∠AOC = 90◦ , ∠ADC = 135◦, 5) ABCD is a square .The diagonals AC and BD intersect at O., , a) What is the measure of angle AOD?, b) What is the measure of angle AP D?, c) What is the measure of angle AQD, Answers, a) Diagonals of a square are perpendicular to eachother.∠AOD = 90◦, b) ∠AP D = 45◦, c) ∠AQD = 180 − 45 = 135◦, , [email protected], 9847307721, , 3, , Prepared by Sri John PA

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2020-21 Academic year Works, Mathematics X, Circles, , 20, Concepts, a) An arc of a circle can make three type of angles. Angle on the arc, angle at the centre and, angle in the complement, b) Angle formed by the arc in the complement is half the angle at the centre, c) Sum of the angle of an arc and its complement is 180◦, , d) Angles on an arc are equal, , Worksheet20, 1) A, B, C, Dare four points on a circle., The chords AC, BDintersect at E.If ∠BEC = 130◦ , ∠ECD = 20◦ then, , a) What is the measure of ∠CED?, b) What is the measure of ∠CDE?, c) What is the measure of ∠BAC?, , a) ∠CED = 180 − ∠BEC, ∠CED = 180 − 130 = 50◦, , Answers, , b) ∠CDE = 180 − (50 + 20) = 180 − 70 = 110◦, c) ∠BAC = ∠BDC, ∠BAC = 110◦, , 1, , Prepared by Sri John PA

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2) O is the centre of the circle.If ∠ACB = 20◦ , ∠CAB = 30◦ then, , a) What is the measure of ∠AOB?, , b) What is the measure of ∠COB ?, c) What is the measure of ∠AOC?, , d) What is the measure of ∠ADC?, , e) What is the measure of ∠ABC?, , Answers, a) Since ∠ACB = 20◦ ,∠AOB = 2 × ∠ACB = 40◦, b) Since ∠CAB = 30◦ ,∠COB = 2 × ∠ACB = 60◦, c) ∠AOC = 40 + 60 = 100◦, d) ∠ADC =, , 1, 2, , × AOC = 50◦, , e) ∠ABC = 150 − 50 = 130◦, 3) Ois the centre of the circumcircle of triangle ABC., If ∠BAC = y, ∠OBC = xthen, , a) What is the measure of ∠BCO?, , b) What is the measure of ∠BOC?, c) Prove that x + y = 90◦, , Answers, a) Since OB = OC opposite angles of these sides in triangle OBC are equal., ∠BCO = x., b) ∠BOC = 180 − 2x, c) We know that ∠BOC = 2 × ∠BAC, 180 − 2x = 2y, 2x + 2y = 180, x + y = 90◦, , 2, , Prepared by Sri John PA

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4) In triangle ABC, ∠A = 70◦ , ∠B = 80◦ . The vertices of the triangle are on the circumcircle of the triangle, .Radius of the circumcircle is 3cm.Construct the triangle., Answers, ⋆ Draw a circle of radius 3 , mark its centre asO and a point A on the circle. Draw the radius, OA, ⋆ Mark a point B on the circle such that ∠AOB = 2 × 70 = 140◦, ⋆ Mark a point C on the circle such that ∠BOC = 2 × 80 = 160◦, ⋆ Draw triangle ABC ., 5) P and Q are the centre of the circles shown in the figure. Circles intersect at B and C.If ∠AQB = 130◦, then, , a) What is the measure of ∠ACB?, b) What is the measure of ∠BCD?, c) What is the measure of ∠BP D, , a) ∠ACB =, , 1, 2, , × 130 = 65, , Answers, , ◦, , b) ∠BCD = 180 − 65 = 115◦, c) The central angle of the complement of the arcBCD is 2 × 115 = 230◦ .Therefore, ∠BP D = 360 − 230 = 130◦, , [email protected], 9847307721, , Prepared by Sri John PA

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2020-21 Academic year Works, Mathematics X, Circles, , 21, Concepts, a) An arc of a circle can make three type of angles. Angle on the arc, angle at the centre and angle in, the complement, b) Angle formed by the arc in the complement is half the angle at the centre, c) Sum of the angle of an arc and its complement is 180◦, d) Angles on an arc are equal, , Worksheet21, 1) In the figure ∠BAC = x, ∠CBO = y, O is the centre of the circle., , a) What is the measure of ∠BCO?, b) What is the measure of ∠BOC?, c) What is the measure of ∠BP C?, , d) Prove that x − y = 90◦ ?, , Answers, , a) ∠BCO = y, (In triangle BCO, OB = OC. Angles opposite to equal sides are equal.), b) ∠BOC = 180 − 2y, c) ∠BP C =, , 1, 2, , × ∠BOC =, , 1, 2, , × (180 − 2y) = 90 − y, , d) ∠BAC + ∠BP C = 180, x + 90 − y = 180, x − y = 180 − 90 = 90◦, , 1, , Prepared by Sri John PA

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2) In the figure O is the centre of the circle .If ∠BCO = 55◦ , ∠BAO = 20◦ then, , a) What is the measure of ∠OBC?, b) What is the measure of ∠BOC?, c) What is the measure of ∠AOC?, d) What is the measure of ∠ABC?, Answers, a) ∠OBC = 55◦, (In triangle OBC, OB = OC. Angles opposite to equal sides are equal.), b) ∠BOC = 180 − (55 + 55) = 180 − 110 = 70◦, c) In triangle AOB,∠B = 20◦ , ∠ACB = 180 − 40 = 140◦, ∠AOC = 140 − 70 = 70◦, d) ∠ABC =, , 1, 2, , × 70 = 35◦, , 3) In the figure O is the centre of the circle.If ∠BAC = 32◦ then, , a) Find the angles of triangle OAB, b) What is the measure of ∠DOC?, c) Find x, Answers, a) In triangle OAB,OA = OB. Therefore ∠B = 32◦ , ∠AOB = 180 − 64 = 116◦, b) ∠DOC = 116◦, (Opposite angles are equal), c) In triangle OCD,∠D = x, x + x + 116 = 180, 2x = 64, x = 32, , 2, , Prepared by Sri John PA

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4) This is the picture of a clock face.1, 8, 5 are joined to make a triangle.Find the angles of this triangle., , Answers, ⋆ The central angle of the arc in between two numbers(say 1, 2) is, The central angle of the arc between 1 and 5 is 4 × 30 = 120◦ ., The angle of the triangle with vertex at 8 is 12 × 120 = 60◦, , 1, 12, , × 360 = 30◦ ., , ⋆ The central angle of the arc between 8 and 5 is 3 × 30 = 90◦ ., The angle of the triangle with vertex at 1 is 12 × 90 = 45◦, ⋆ The central angle of the arc between 8 and 1 is 5 × 30 = 150◦ ., The angle of the triangle with vertex at 5 is 12 × 150 = 75◦, , 5) Angles of a triangle are in the ratio1 : 2 : 3. Vertices of this triangle are on a circle of radius 3cm .Construct, the triangle, Answers, Draw a circle of radius 3cm, Divide the angle around the centre in the ratio 1 : 2 : 3., The resulting angles are 60◦ , 120◦ , 180◦ by drawing radii. Join the ends of the radii on the circle ., , [email protected], 9847307721, , 3, , Prepared by Sri John PA

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2020-21 Academic year Works, Mathematics X, Circles, , 23, Concepts, a) If the vertices of a qudrilateral are on a circle we call it cyclic quadrilateral., b) The sum of the opposite angles of a cyclic quadrilateral is 180◦ ., c) The converse of the above statement is also true. If the sum of the opposite angles of a quadrilateral, is 180◦ it will be a cyclic quadrilateral., d) Square, rectangle and isosceles trapezium are cyclic ., Worksheet23, 1 In the figure BD = CD,∠DBC = 25, , ◦, , a) What is the measure of ∠BDC?, b) What is the measure of ∠BAC?, c) What is the measure of ∠EBC?, Answers, a) In triangle BDC, BD = CD. Angle opposite to these sides are equal .∠BCD = 25◦, ∠BDC = 180 − (25 + 25) = 130◦, b) ∠BAC = 180 − 130 = 50◦, c) ∠BEC = ∠BAC = 180−130 = 50◦ , ∠EBC = 180−(90+50) = 180−140 = 40◦, 2) Two circles intersect at B and E as in the figure. The points A − B − C are along a line. Also the points, D − E − F are also on a line., , 1, , Prepared by Sri John PA

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a) Prove that AD is parallel to CF, b) If AC = DF suggest a suitable name to the quadrilateral ADF C, c) Prove that ADF C is a cyclic quadrilateral., Answers, , a) Draw BE .ABED is a cyclic quadrilateral., If ∠DAB = x then ∠BED = 180 − x,∠BEF = 180 − (180 − x) = x., BEF Cis cyclic .∠C = 180 − x., In quadrilateral ADF C, ∠A + ∠C = x + 180 − x = 180◦, Co interior angle sum is 180◦ . ADis parallel to CF, b) ADF Cis a trapezium .Since AC = DF is an isosceles trapezium ., c) Angles at the ends of parallel sides of an isosceles trapezium are equal .Since ∠A = ∠D, and ∠A + ∠C = 180◦ then ∠D + ∠C = 180◦ ., ADF Cis a cyclic quadrilateral., 3) AB is the diametre of the circle.CD is a chord of length equal to radius of the circle., , a) What is the measure of ∠COD?, b) What is the measure of ∠CBD?, c) What is the measure of ∠DCP ?, d) Find the measure of ∠CP D, , 2, , Prepared by Sri John PA

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Answers, a) Draw OC, OD ,OCD is an equilateral traingle .∠COD = 60◦, b) ∠CBD =, , 1, 2, , × 60 = 30◦, , c) ∠BCP = 90◦ (angle in the semicircle ). ∴ BCP = 90◦ ., d) In traingle BCP , ∠CP D = ‘∠CP B = 180 − (90 + 30) = 60◦, 4) In the figure ABCD is a quadrilateral in which AB is parallel to CD and AD = BC, , Prove that ABCD is a cyclic quadrilateral., Answers, ⋆ The line AB is parallel to CD. Therefore ∠A + ∠D = 180◦, ⋆ Since AD = BCthen ABCD is an isosceles trapezium ∠A = ∠B, ⋆ Therefore ∠B + ∠D = 180◦ . ABCDis a cyclic quadrilateral ., 5) The angles of the quadrilateral ABCD are in the ratio 1 : 2 : 3 : 4 in an order., a) If the smallest angle is x,what are the other angles?, b) Find the measure of all the angles of ABCD, c) Is ABCD a cyclic quadrilateral., d) How should the ratio numbers interchange to make this cyclic?, Answers, a) Angles are x, 2x, 3x, 4x, b) x + 2x + 3x + 4x = 360, 10x = 360, x = 36, angles are 36◦ , 72◦ , 108◦ , 144◦, ∠A + ∠C = 36 + 108 ̸= 180.This is not cyclic ., c) If the ratio is changed into1 : 2 : 4 : 3,opposite angle sum becomes 180◦ . This is a cyclic, quadrialteral, , [email protected], 9847307721, , 3, , Prepared by Sri John PA

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2020-21 Academic year Works, Mathematics X, Circles, , 19, Concepts, a) An arc of a circle can make three type of angles. Angle on the arc, angle at the centre and angle in, the complement, b) Angle formed by the arc in the complement is half the angle at the centre, c) Sum of the angle of an arc and its complement is 180◦, d) Angles on an arc are equal, Worksheet19, 1) OABC is a parallelogram.Three vertices are on a circle and one at the centre.P is a point on the circle, , a), b), c), d), , Draw AP and CP , mark the angleAP C = x. What is ∠AOC, What is angle ABC?, Find x, Find the angles of the parallelogram, Answers, , a) ∠AOC = 2x, , b) ∠ABC = 2x Opposite angles of a parallelogram are equal, c) ∠AP C + ∠ABC = 180◦ ,x + 2x = 180, 3x = 180, x = 60, d) Angles are 120◦ , 60◦ , 120◦ , 60◦, 2) In the figure O is the centre of the circle,∠BAO = 20◦ , ∠BCO = 10◦, , 1, , Prepared by Sri John PA

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a) What is the measure of angleABC?, b) What is the measure of angleAOC?, c) What is the measure of angleADC?, d) Find the angles of triangle AOC, e) If the diametre of the circle is 10cm then find the length of the chord AB, Answers, a) In triangle OAB,OA = OB. Angles opposite to the equal sides are equal.Similarly in the, case of triangle OBCalso., ∠ABC = 20 + 10 = 30◦, b) ∠AOC = 2 × 30 = 60◦, c) ∠ADC = 180 − 30 = 150◦, d) Triangle AOC, OA = OC, ∠OAC = ∠OCA =, triangle. Angles are 60◦ each., , 180−60, 2, , = 60◦ △OAC is an equilateral, , e) OA = AC = OC = 5cm, radius 5cm., 3) In the figure O is the centre of the circle.If angle ADC = 140◦ , angleAEC = 60◦ then, , a) What is the measeure of ∠AP C and ∠AQC, b) What is the measure of angle AOC?, c) Fnd the angles of the quadrlateralP EQB, Answers, a) ∠AP C = 180 − 140 = 40 , ∠AQC = 40◦, ◦, , b) ∠AOC = 2 × 40 = 80◦, c) In the quadrilateral ∠AEQ = ∠AEC = 60◦ , ∠EP B = 180 − 40 = 140◦ , ∠EQB =, 140◦, ∠P BQ = 360 − (140 + 140 + 60) = 20. Angles are 140◦ , 60◦ , 140◦ , 20◦, 4) In the figure Ois the centre of the circle, ∠OAC = 45◦ then, , 2, , Prepared by Sri John PA

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a) What kind of triangle is OAC?, b) What is the measure of angleABC?, c) What is the measure of angle ADC?, d) If the radius of the circle is 6cm then what is the length of the chord AC., Answers, a) OA = OC, ∠OAC = ∠OCA = 45◦ ,∠AOC = 90◦ . △OAC is an isosceles right, triangle, b) ∠ABC = 12 AOC = 45◦, c) ∠ADC = 180 − 45 = 135◦, √, √, d) AC = 62 + 62 = 6 2cm, 5) Draw a circle of radius 3cm , construct an equilateral triangle with vertices on the circle. What is the length, of the side?, Answers, ⋆ Draw a circle with centre O and radius 3cm. Mark a point A on the circle and radiusOA., ⋆ Divide the angle around the centre 120◦ each and mark the points B, Con the circle, ⋆ Draw triangle ABC ., √, ⋆ Length of side = 3 3cm, , [email protected], 9847307721, , 3, , Prepared by Sri John PA

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2020-21 Academic year Works, Mathematics X, Circles, , 22, Concepts, a) If the vertices of a qudrilateral are on a circle we call it cyclic quadrilateral., b) The sum of the opposite angles of a cyclic quadrilateral is 180◦ ., c) The converse of the above statement is also true. If the sum of the opposite angles of a quadrilateral, is 180◦ it will be a cyclic quadrilateral., d) Square, rectangle and isosceles trapezium are cyclic ., , Worksheet22, 1) In the figure P QRS is a cyclic quadrilateral.∠P = x, ∠Q = y, ∠R = 3x, ∠S = 5y., , a) Find x and y, b) Find the angles of the quadrilateral., ◦, , Answers, , a) ∠P + ∠R = 180, x + 3x = 180, 4x = 180, x = 45, ∠Q + ∠S = 180◦ , y + 5y = 180, 6y = 180, y = 30, b) ∠P = 45◦ , ∠R = 3 × 45 = 135◦ ∠Q = 30◦ , ∠S = 5 × 30 = 150◦, , 1, , Prepared by Sri John PA

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2) ABC is an isosceles triangle with AB = AC, ∠ABC = 50◦ ., , a) Name two cyclic quadrilaterals in this picture., b) What is the measure of angle D?, c) What is the measure of ∠BEC?, Answers, a) Quadrilateral ABEC and quadrilateral DBEC are cyclic., b) ∠ABC = ∠ACB = 50◦, ∴ ∠A = 180 − 100 = 80◦, ∴ ∠D = 80◦, c) ∠BEC = 180 − 80 = 100◦, 3) ABCD is a cyclic quadrilateral.AB is the diametre of the circle , AD = CD and ∠ADC = 130◦ ., , a) What is the measure of ∠ACB?, b) What is the measure of ∠ABC?, c) Find ∠DCB., d) What is the measure of ∠BAD?, Answers, a) ∠ACB = 90 (Angle in the semicircle), ◦, , b) ∠ABC = 180 − 130 = 50◦, c) Since CD = AD,the angles opposite to the equal sides of triangle ADCare equal., ∠DCA = 25◦ , ∠DCB = 90 + 25 = 115◦, d) ∠BAD = 180 − 115 = 65◦, , 2, , Prepared by Sri John PA

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4) Prove that any cyclic parallelogram is a rectangle., Answers, ⋆ ABCD is a parallelogram .(Draw rough figure), ⋆ Opposite angles are equal. ∠A = ∠C, ∠B = ∠D, ⋆ Sum of the opposite angles is 180◦, ⋆ ∠A + ∠C = 180◦ , ∠A = ∠C ∴ ∠A = 90◦ , ∠C = 90◦, ⋆ ∠B + ∠D = 180◦ , ∠B = ∠D∴ ∠B = 90◦ , ∠D = 90◦, ⋆ ABCDis a square, 5) In triangle ABC, AB = AC.P and Q are the mid points of the side AB and AC., a) Draw a rough diagram and join the points P and Q., b) Prove that BP QC is a cyclic quadrilateral., , c) If ∠A in triangle ABC is 20◦ , find the angles of the trapezium BP QC, Answers, a) Figure, , b) Since AB = AC,∠B = ∠C., Line joining the mid points of two sides of a triangle is parallel to the third side. P Q is, parallel to BC., In P BCQ, ∠B + ∠P = 180◦ (co interior angles ), Since ∠C = ∠B∠C + ∠P = 180◦, P QCB is a cyclic quadrilateral, c) ∠A = 20◦, ∠B = ∠C = 180−20, = 80◦, 2, ◦, ∠B + ∠P = 180 , ∠P = 100◦ ,∠Q = 100◦, Angles are ∠P = 100◦ , ∠Q = 100◦ , ∠B = 80◦ , ∠C = 80◦, , [email protected], 9847307721, , 3, , Prepared by Sri John PA

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2020-21 Academic year Works, Mathematics X, Circles, , 23, Concepts, a) If the vertices of a qudrilateral are on a circle we call it cyclic quadrilateral., b) The sum of the opposite angles of a cyclic quadrilateral is 180◦ ., c) The converse of the above statement is also true. If the sum of the opposite angles of a quadrilateral, is 180◦ it will be a cyclic quadrilateral., d) Square, rectangle and isosceles trapezium are cyclic ., , Worksheet23, 1) C, D are two points in a semicircle of diametre AB., If ∠BAD = 70◦ , ∠DBC = 30◦ then, , a) What is the measure of ∠BCD?, b) What is the measure of ∠CDB?, c) What is the measure of ∠ADC?, d) What is the measure of ∠ABD?, , a) ∠BCD = 180 − 70 = 110, , ◦, , Answers, , b) ∠CDB = 180 − (30 + 110) = 180 − 140 = 40◦, c) ∠ADC = ∠ADB + ∠BDC = 90 + 40 = 130◦, d) ∠ABD = 180 − (90 + 70) = 180 − 160 = 20◦, , 1, , Prepared by Sri John PA

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2) In the figure we can see three intersecting circles .D − P − Q − Care on a line .A − R − S − Bare also, on a line, , a) Prove that the quadrilateral ABCDis cyclic, b) If ∠CDA = ∠DAB then what type of quadrilateral is ABCD?, , c) If ∠CDA = ∠DAB = 40◦ then find other two angles of ABCD, Answers, a) Join P R and QS in the figure., DARP is a cyclic quadrilateral., If ∠ADP = x,then ∠ARP = 180 − x,∠P RS = x., Since PRSQ is cyclic, ∠P QS = 180 − x,∠SQC = x, ∠SBC = 180 − x., In the quadrilateral ABCD,∠D + ∠B = x + 180 − x = 180◦, ABCD is cyclic., b) Quadrilateral ABCD will be an isosceles trapezium., c) ∠ABC = 140◦ , ∠DCB = 140◦, , 3 In traingle ABC, P is a point on BC., AB = AP ,the line through A parallel to BC and the line through Cparallel to AP intersect at D .Prove, that ABCD is cyclic, , Answers, a) In traingle ABP ,AB = AP . Angles opposite to the equal sides are equal., ∠ABP = x then ∠AP B = x,∠AP C = 180 − x., Since AP CD is a parallelogram ∠D = 180 − x., ∠B + ∠D = x + 180 − x = 180◦ ., ABCD is cyclic, 4) The parallelogram which is not a rectangle is not cyclic. Justify this statement, Answers, ⋆ ABCD is a parallelogram(Draw a parallelogram and name it as ABCD in an order., ∠A = ∠C, ⋆ Since ABCD is not a rectangle ∠A =, ̸ 90◦ , ∠C ̸= 90◦, ⋆ ∠A + ∠C =, ̸ 180◦ ., ∴ ABCD is not cyclic, , 2, , Prepared by Sri John PA

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5) A, B, C, D, P, Q, R, Sare the points on a circle Find ∠P + ∠Q + ∠R + ∠S, , Answers, , a) Draw SC and SD., ∠SBP C is cyclic.∠1 + ∠P = 180◦, b) SCQD is cyclic.∴ ∠2 + ∠Q = 180◦, c) SDRA is cyclic. ∴ ∠3 + ∠R = 180◦ ., d) Adding,, ∠1 + ∠2 + ∠3 + ∠P + ∠Q + ∠R = 3 × 180, ∠S + ∠P + ∠Q + ∠R = 540◦, , [email protected], 9847307721, , 3, , Prepared by Sri John PA

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2020-21 Academic year Works, Mathematics X, Circles, , 24, Concepts, a) If the vertices of a qudrilateral are on a circle we call it cyclic quadrilateral., b) The sum of the opposite angles of a cyclic quadrilateral is 180◦ ., c) The converse of the above statement is also true. If the sum of the opposite angles of a quadrilateral, is 180◦ it will be a cyclic quadrilateral., d) Square, rectangle and isosceles trapezium are cyclic ., Worksheet24, 1) In the figure O is the centre of the circle, ∠DAB = 50◦, , a), b), c), d), , Find x, Find y, If BC = CDthen what is the measure of ∠ADC?, If BC = CDthen what is the measure of ∠ABC?, a) x = 2 × 50 = 100, , Answers, , ◦, , b) ABCDis a cyclic quadrilateral .y + 50 = 180 ,y = 180 − 50 = 130, c) Draw BD.In traingle BDC, CD = CB, Angles opposite to the equal sides are equal ., ∠CDB = ∠DBC = 180−130, = 25◦, 2, In triangle ODB, OD = OB.Angles opposite to the equal sides are equal.∠ODB =, ∠OBD = 180−100, = 40◦, 2, ∠ADC = 25 + 40 = 65◦, d) ∠ABD = 90◦ , ∠DBC = 25◦, ∠ABC = 90 + 25 = 115◦, , 1, , Prepared by Sri John PA

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2) In the figure ∠BAC = 60◦ ,∠BCA = 20◦, , a) Looking into the figure Riswan said: ACis the diametre of the circle .Can you agree with his opinion?, Why?, b) What is the measure ∠ADC, c) If ∠DAC : ∠DCA = 3 : 1 then find these angles., Answers, , a) ∠B = 180 − (60 + 20) = 100◦ ̸= 90◦, AC is not the diametre, , b) ABCDis cyclic .∠ADC = 180 − 100 = 80◦, c) In triangle ACD, ∠A + ∠C = 180 − 80 = 100◦ ., Divide 100 in the ratio 3 : 1, ∠A = 75◦ , ∠C = 25◦, 3 In the figure ABCDEis a regular pentagon.Prove that ABCEis a cyclic quadrilateral., , Answers, ⋆ One angle of the regular pentagon is =, , (5−2)×180, 5, , = 108◦, , ⋆ In triangle EDC, ED = CD, ∠DEC = ∠DCE =, , 180−108, 2, , = 36◦, , ⋆ ∠ECB = 108 − 36 = 72◦ . In the quadrilateral ABCE, ∠A + ∠C = 108 + 72 = 180◦ ., ABCEis cyclic., 4) Prove that the trapezium having diagonals equal is cyclic, , 2, , Prepared by Sri John PA

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Answers, ⋆ In the figure ABCD is a trapezium having equal diagonals .ABis parallel to CD ,also, AC = BD, ⋆ DrawDP and , CQ perpendicular to AB.We can see traingles DP B and CQA.These are, right angled triangles .Hypotenues are equal . Other two sides are equal (DP = CQ).These, triangles are equal . ∠CAQ = ∠DBP, ⋆ ∠CAQ = ∠DBP are the angles of triangle ACB and triangle BDA., Consider triangle ACB, and triangle BDA, AB is the common side , AC = BD, including, angles are equal .These triangles are equal, ⋆ Base angles of trapezium ABCDare equal . This is an isosceles trapezium . Since ∠A =, ∠B, ∠A + ∠D = 180◦ , ∠A + ∠C = 180◦ . ABCDis cyclic, 5) ABCDis a cyclic quadrilateral. If ∠A − ∠C = 60◦ then find the measure of ∠C.What is the measure of, ∠A?, Answers, ⋆ ∠A − ∠C = 60◦ , ∠A + ∠C = 180◦ ., ⋆ Adding these equations ,2 × ∠A = 240, ∠A = 120◦ , ∠C = 180 − 120 = 60◦, , [email protected], 9847307721, , 3, , Prepared by Sri John PA

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2020-21 Academic year Works, Mathematics X, Circles, , 25, , Concepts, a) Two chords of a circle AB and CD intersect at the point P inside, the circle .It can be proved that P A × P B = P C × P D, b) This relation can be used to construct a rectangle having equal area, of another rectangle., c) If the chords intersect outside the circle ,the same relation holds., PA × PB = PC × PD, Worksheet 25, 1) In the figure two chords AB and CD intersect inside a circle at P ., , ., a) Join AC and BD. Establish the similarity of triangle P AC and P BD, b) What are the equal angles of these triangles, c) Prove that P A × P B = P C × P D, , 1, , Prepared by Sri John PA

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Answers, , a) ∠ACD = ∠ABD, ∠CAB = ∠CDB(Angle in the same arc, are equal), In the triangles ACP and DBP, ∠ACP = ∠DBP, ∠CAP = ∠P DB, These triangles are similar., b) ∠C = ∠B, ∠A = ∠D, A, c) Sides opposite to the equal angles are proportional. PP D, =, PA × PB = PC × PD, , 2) In the figure the chord AB has length 8cm and OA = 5cm., , a) What is the length of OB?, b) IfOC = 2.5cm, what is the length of OD?, , 2, , Prepared by Sri John PA, , PC, PB

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Answers, , a) OB = 8 − 5 = 3cm, b) OA × OB = OC × OD, 5 × 3 = 2.5 × OD, OD =, , 15, 2.5, , = 6cm, , 3) In the figure AB = 5cm, BD = 4cm, CD = 9cm., , ., a), b), c), d), , What is the length of AD?, Calculate the length of DE?, Is CE the diameter of the circle? why?, Find the length of CE, a) AD =, , √, , Answers, 52 − 42 = 3cm, , b) DA × DE = CD × DB, 3 × DE = 9 × 4, DE = 12cm, c) CE is not the diametre . If it is a diametre ,∠D = 90◦ has the, vertex on the circle, √, d) CE = 122 + 92 = 15cm, 3, , Prepared by Sri John PA

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4) If AB and CD are two chords of a circle which when produced meet at a, point P . If P A = P C show that AB = CD., , Answers, ⋆ PA × PB = PC × PD, Cancel P A and P C on both sides., P B = P D, P A − AB = P C − CD, Again cancel P A and P C on both sides . −AB = −CD,, AB = CD, 5) In the figure AB and CD are two chords of a circle which when produced, meet at a point P, a) Draw AC and BD , complete the quadrilateral ABDC, b) Establish the similarity of the triangles P AC and P DB, c) Establish the relation P A × P B = P C × P D, , 4, , Prepared by Sri John PA

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Answers, , ⋆ ABDC is a cyclic quadrilateral, If ∠C = x, ∠B = 180 − x and ∠P BD = x., ∠ACP = ∠DBP, ⋆ If ∠A = y, ∠BDC = 180 − y and ∠BDP = y., ∠CAP = ∠BDP, ⋆ Triangles P CA and P BD are similar.Sides opposite to the, equal angles are proportional., PA, PC, PD = PB, PA × PB = PC × PD, [email protected], 9847307721, , 5, , Prepared by Sri John PA

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2020-21 Academic year Works, Mathematics X, Circles, , 25, Concepts, a) Two chords of a circle AB and CD intersect at the point P inside the circle .It can be proved that, PA × PB = PC × PD, b) This relation can be used to construct a rectangle having equal area of another rectangle., c) If the chords intersect outside the circle ,the same relation holds. P A × P B = P C × P D, , Worksheet 25, 1) The chords AB and CD intersect at O .This point divide each chord into two segments, , a) What is the relation between these segments?, b) If CD = 10cm and OD = 4cm then what is the length OC?, c) If OA = 8cm , OC = 6cm and OD = 4cm then what is the length OB?, Answers, , a) OA × OB = OC × OD, , b) OC = CD − OD = 10 − 4 = 6 cm, c) 8 × OB = 6 × 4, OB = 3cm, , 1, , Prepared by Sri John PA

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2) The chords AB and CD intersect at P outside the circle., , a) What is the relation between P A, P B, P C and P D?, b) If AB = 5cm , P B = 3cm , P D = 2cm then what is the length CD?, Answers, , a) P A × P B = P C × P D, b) (5 + 3) × 3 = (P D + CD) × P D, (5 + 3) × 3 = (2 + CD) × 2, 2 + CD = 12, CD = 10cm, , 3) In the trapezium ABCD, AD = BC and AB is parallel to CD . The diagonals AC and BD intersect at, P., , a) What is the relation between ∠ADB and ∠ACB? How can we realize this relation?, , b) If ∠DAC = 30◦ then what is the measure of ∠DBC?, , c) What is the relation between the segments made byP on the diagonals?, Answers, a) ABCD is an isosceles trapezium.Therefore it is a cyclic trapezium., A circle passes through A, B, C, D.In this circle the arc AB makes two angles, ∠ADB, ∠ACDin its complement .These are equal, ∠ADB = ∠ACD, b) The arc CD makes two angles ∠DAC, ∠DBCin its complement .These are equal, Since ∠DAC = 30◦ ,∠DBC = 30◦, c) AC ം BD are the diagonals .These are the chords of the circle passing through the vertices, . These chods meet at P ., PA × PC = PB × PD, 4) In the quadrlateral ABCD , the diagonals AC and BD intersect at P . If P A = 9cm, P B = 12cm, P C = 4cm and P D = 3cm then, a) Draw a rough diagram and mark the mesaurements, b) Is this a cyclic quadrilateral ? How can we realize this ?, , 2, , Prepared by Sri John PA

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c) If ∠A = 40◦ and ∠B = 70◦ find other two angles of the quadrilateral, Answers, a) Picture, , b) P A × P C = 9 × 4 = 36, P B × P D = 12 × 3 = 36, PA × PC = PB × PD, There will a circle with the diagonals as the chords .ABCDis a cyclic quadrilateral., c) ∠A = 40◦ ∴ ∠C = 180 − 40 = 140◦, ∠B = 70◦ ∴ ∠D = 180 − 70 = 110◦, 5) Draw a rectangle of sides 4cm and 6cm . Construct another rectangle with area equal to the area of the, first rectangle and one side 7cm in length., Answers, ⋆ Draw a circle with the sides AB = 6 cm and AD = 4cm . Produce AB to E , 1 more such, that AE = 7, ⋆ Draw a arc with centre A and radius AE.This arc intresect DA produced at F .Produce BA, such that AD = AG, ⋆ Draw circumcircle of triangle BF G. This circle intersect AD at H.Draw a rectangle with, sides AE and AH.Area of this rectangle will be equal to the area of the rectangle ABCD., Note :The first rectangle have area AG × AB.AE × AHis the area of second rectangle ., These areas are equal because AB × AG = AF × AH, , [email protected], 9847307721, , 3, , Prepared by Sri John PA

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2020-21 Academic year Works, Mathematics X, Circles, , 26, Concepts, , a) In the case of the intersecting chords of a circle,if one chord AB is the diametre of the circle and, other chordCD is perpendicular to the diametre ,then P A × P B = P C 2, b) This relation is used to construct a square with same area of a rectangle.It can be used to draw the, lines of irrational lengths., Worksheet 26, 1) AB is the diametre of a semicircle, P is a point on AB and P C is perpendicular to AB, , a) Prove that P A × P B = P C 2, b) IfP A = 9cm , P B = 4 cm then what is the length P C?, c) What is the area of the square with side P C?, Answers, a) Draw AC, BC.Consider triangle AP C and triangle BP C.If∠P AC = xthen ∠P CA =, 90 − x, ∠P CB = 90 − (90 − x) = x, ∠P BC = 90 − x, PC, PA, These are similar triangles .Sides opposite to the equal angles are proportional . P, B = PC, 2, PA × PB = PC ., b) P C 2 = 9 × 4 = 36, P C = 6cm, c) Area P C 2 = 36sq.cm, , 1, , Prepared by Sri John PA

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2) ]AB is the diametre of a semicircle, P is a point on AB and P C is perpendicular to AB, , a) If P C = 6cm,and P B = 3cm then what is the length of P A, b) What is the radius of the circle ?, c) What is the area of the square drawn with side P C?, Answers, a) P A × P B = P C, P A × 3 = 62 , P A = 12cm, 2, , b) AB = 12 + 3 = 15cm ,Radius 7.5cm., c) Area P C 2 = 36sq.cm, √, 3) In the figure AB is the diametre of the semicircle, P C is perpendicular to AB. AC = 5 29cm and, P A = 25cm., , a) What is the length of P C?, b) What is the lenght P B?, c) What is the radius of the circle?, √ √, a) P C = (5 29)2 − 252 = 10 cm, , Answers, , b) P A × P B = P C 2 ,25 × P B = 102 , P C = 4cm, c) AB = 25 + 4 = 29, Radius= 14.5cm, √, 4) Draw a semicircle of suitable diametre .Construct a line of length 12cm perpendicular to the diametre, whose one end is on the diameter and other end is on the semicircle.Explain the principle of construction., , 2, , Prepared by Sri John PA

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Answers, a) Take two positive integers having the product 12.Draw a circle with the sum of these numbers, as the diametre., 12 = 6 × 2, diametre = 6 + 2 = 8cm., Draw a line AB of lenght 8cm , and a semicircle with AB as the diametre., b) Mark a point P at the diatance 6 cm from A. Draw a perpendicular to AB at P .This line cut, the semicircle at C, c) P C 2 =√P A × P B = 6 × 2 = 12, P C = 12െസ ീമീ ർ, 5) In the figure AB is the diametre of the circle and P C is perpendicular to the diametre. P A : P B = 2 : 1, and P C = 6cm., , a) Write the relation between P A, P B and P C?, b) Find the lengths P A and P B, c) What is the radius of the circle?, Answers, a) P A × P B = P C, , 2, , b) If P B =√, x ,2x × x =√62 ,2x2 = 36, x2 = 18, x =, P A = 6 2, P B = 3 2, √, √, √, c) AB = 6 2√+ 3 2 = 9 2, Radius = 9 2 2 cm, , √, , 18 =, , √, √, 9×2=3 2, , [email protected], 9847307721, , 3, , Prepared by Sri John PA

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2020-21 Academic year Works, Mathematics X, Circles, , 26, Concepts, , a) In the case of the intersecting chords of a circle,if one chord AB is the diametre of the circle and, other chordCD is perpendicular to the diametre ,then P A × P B = P C 2, b) This relation is used to construct a square with same area of a rectangle.It can be used to draw the, lines of irrational lengths., Worksheet 26, 1) AB is the diametre of a semicircle.The lines P Qand RSare perpendicular to AB.If P Q = RSthen, , a) What is the relation between the lengths P A, P B and P Q ?, b) What is the relation between the lenghths AR, BR and RS, c) Prove that P A = BR, Answers, a) P A × P B = P Q, , 2, , b) AR × BR = RS 2, c) Since RS = P Q ,RS 2 = P Q2, P A × P B = AR × BR, P A(P R + BR) = BR(P A + P R), P A × P R + P A × BR = P A × BR, 1 + BR × P R, Cancel P A × BRfrom both sides, P A × P R = P R × BR, P A = BR, , Prepared by Sri John PA

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2), , a) Draw an equilateral triangle of altitude 3 cm, b) What is the lenght of one side ?, c) What is the radius of its incircle?, Answers, a) Draw a line AD of length 4cm. Draw a circle with diametre AD.Mark a point P at a distance, 3cm from A. From P draw a line perpendicular toAD. This line cut the circle at C, B .Draw, triangle ABC., √, √, P C 2 =√, 3 × 1, P C = 3, BC = 2 3cm, √ 2 √, √, AC = 32 + 3 = 12 = 2 3cm., √, √, AB = 2 3cm . AB = BC = AC = 2 3cm, √, b) AB = AC = BC = 2 3, , 3) Draw a rectangle of sides 5cm and 3cm .Construct a square whose area is same as the area of the rectangle, Answers, ⋆ Draw the quadrilateral ABCD. AB = 5cm , BC = 3cm., ⋆ Produce AB and mark the point E such that BC = BE, ⋆ Draw a semicircle of diametre AE.Produce BC, meet the semixircle at F ., ⋆ BA × BE = BF 2 can be written as AB × BC = BF 2 . AB × BC is the area of the, rectangle, ⋆ Draw a square of side BF .Area of rectangle is equal to the area of the square as per the, relation AB × BC = BF 2, 4), , √, a) Draw a semicircle of suitable diametre .Draw a line of length 12cm whose one end on AB and other, end on the semicircle., √, b) Draw a chord of length 48 cm by make the semicircle as the circle, Answers, ⋆ Take two numbers with product 12, draw a circle with diameter as the sum of these numbers, .12 = 6 × 2, diametre = 6 + 2 = 8cm ., Draw a line of AB = 8cm. Draw a circle with this line as the diametre, Mark a point P at the diatance 6cm from A, draw a perpendicular line from P to the diametre, .This line cut the semicircle at C., P C 2 =√P A × P B = 6 × 2 = 12, P C = 12cm, ⋆ Make semicircle, as, √ the circle . √Produce CP in such a way as the meet the circle at D., √, CD = 2 12 = 22 × 12 = 48cm, , 5) ABis the diametre of a semicircle.P Q =, diametre . Find the length of AB?, , √, √, 14cm RS = 18 cm . These lines are perpendicular to the, , Answers, √, √, ⋆ P Q = 14 = 7 × 2, Length of AP is 2cm . length of BP cm 7 ., Then AB = 9cm., √, √, ⋆ RS = 18 = 6 × 3, Length of ARis 6cm ,length of BRis 3cm ., Then AB = 9cm., ⋆ In both cases AB = 6 + 3 = 9 and AB = 2 + 7 = 9. AB = 9 cm, , 2, , Prepared by Sri John PA

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6) AB is the diametre of a semicircle , P Q is parallel to the diametre, If AB = 8cm , BQ = 2 cm then find the langth P Q., , Answers, , ⋆ Draw a perpendicular from Q to AB. Draw QC, If CB = x then CA = 8 − x, ⋆ CB × CA = CQ2, x(8 − x) = CQ2, ⋆ In the right triangle QCB , CQ2 = 22 − x2 = 4 − x2, ⋆ x(8 − x) = 4 − x2 , 8x − x2 = 4 − x2 , 8x = 4, x =, , 4, 8, , =, , 1, 2, , ⋆ Draw a perpendicular from P to AB ,, ., AD = 12 ആ, P Q = 8 − ( 12 + 12 ) = 7 cm, 7) Draw an equilateral triangle of one side, , √, 18cm, , ⋆ 18 = 6 × 3, 6 + 3 = 9, Draw a line ABof length 9cm, ⋆ Draw a semicircle with diameter AB. Mark a point P at the diatance, 6cm from A. Draw a, √, perpendicular from P to AB. This line cut the circle at C. P C = 18., ⋆ Draw an equilateral triangle with P C as side, , [email protected], 9847307721, , 3, , Prepared by Sri John PA