Page 2 :
Class-10, Mathematics part-2, Question bank, 1.Similarity, Q.1 A) MCQ ( 1 Mark), 1.If ∆ABC~ ∆PQR and AB: PQ = 3: 4 then A(∆ABC): A(∆PQR) =?, (A)9:25, , (B) 9:16, , (C) 16:9, , (D)25:9, , 2.Which of the following is not a test of similarity?, (A)AAA, , (B)SAS, , (C) SAA, , (D)SSS, , 3.If ∆XYZ ~ ∆PQR and A(∆XYZ ) = 25 𝑐𝑚2 , A(∆PQR) = 4 𝑐𝑚2 then XY: PQ =?, (A) 4:25, , (B)2:5, , (C) 5:2, , (D)25:4, , 4.Ratio of areas of two similar tringles is 9:25. _____ is the ratio of their, corresponding sides., (A)3:4, , (B)3 :5, , (C) 5:3, , (D)25:81, , 5. 𝐺𝑖𝑣𝑒𝑛 ∆ABC~ ∆DEF, if ⦟A = 45° and ⦟E = 35° then ⦟B =?, (A) 45°, , (B)35°, , (C)25°, , 6. In fig,seg DE ⃦ seg BC, identify correct statement., (A)DB =AC, , AD, , AE, , (B)DB =AC, , AD, , EC, , (D) DB =EC, , (C) DB =AC, , 7.If ∆XYZ~ ∆PQR then, , XY, , =, , AD, , AB, , AD, , AE, , YZ, , PQ QR, , =?, , (D) 40°
Page 3 :
XZ, , XZ, , (A) PR, , YZ, , (C) QR, , (D) PQ, , If ∆ABC~ ∆LMN and ⦟A = 60° then ⦟L =?, , 8., (A), , XZ, , (B)PQ, , 45°, 9. In, , (B)60°, ∆DEF and ∆XYZ ,, , (C)25°, DE, , FE, , =, XY YZ, , (D) 40°, , & ⦟E ≅ ⦟Y ______ test gives similarity, , between ∆DEF & ∆XYZ., (A)AAA (B)SAS, (C) SAA (D)SSS, , 10. In fig BD=8, BC=12, , Q.1 B), , B-D-C then, , (A)2:3, , (B)3:2, , (C) 5:3, , (D)3:4, , Solve, , A(∆ABC), =?, A(∆ABD), , 1 mark, , B.1 Are triangles in figure similar ? If yes then write the test of similarity., , 2. In fig line BC ⃦ line DE, AB=2 ,BD=3 ,AC=4 and CE= x , then find the value of x.
Page 4 :
3.State whether the following triangles are similar or not : If yes , then write the test of, similarity., P, , x, , Y, 60°, , Q, , ⦟P = 35° , ⦟x = 35° and ⦟Q = 60°, ⦟Y =, , Z, , R, , 4. If ∆ABC~ ∆LMN & ⦟B = 40° then ⦟M =? Give reason ., 5.Areas of two simlar triangles are in the ratio 144:49. Find the ratio of their, corresponding sides., 6. ∆PQR~ ∆SUV write pair of congruent angle., 7. ∆ABC~ ∆DEF write ratio of their corresponding sides., 8., , R, , In fig. TP =10 cm PS=6 cm, , A∆(RTP), A(∆RPS), , T, , P, , =?, , S, , 9.Ratio of corresponding sides of two similar triangles is 4:7 then find the ratio of, their areas = ?
Page 5 :
10. Write the test of similarity for triangles given in figure., , Q.2 A.Complete the activity 2marks, 1., , A, , in fig. BP, , P, , Q, , AC,CQ, , AB A-P-C, , & A-Q-B then, , show that, , ∆APB & ∆AQC are similar, B, , C, , In, , ∆APB & ∆AQC, , ⦟APB = [ ]0 … (𝐼), , ⦟AQC = [ ]0 … (𝐼𝐼), ⦟APB ≅ ⦟AQC (I) & (II), ⦟PAB ≅ ⦟QAC [...........], ∆APB~ ∆AQC [..........], 2.Observe the figure & complete following activity., in fig⦟B = 750 , ⦟D = 750, ⦟B ≅ [… . ] each of 750, ⦟C ≅ ⦟C, , [....], , ∆ABC~ ∆[........], ....[.....]similarity test, , 3., , ∆ABC~ ∆PQR , A( ∆ABC)= 80sqcm A(∆PQR) = 125 sqcm, , then complete
Page 6 :
A(∆ABC), A( ∆PQR), , 80, , [….], , = 125 = [….] hence, , AB, , […..], , =, , PQ […..], , 4.in fig.PM=10 cm A( ∆PQS)= 100sqcm A( ∆QRS) = 110sqcm then NR?, ∆PQS &∆QRS having seg QS common base, Areas of two triangles whose base are common, are in proportion of, their corresponding [.......], A(∆PQS), A( ∆QRS), , Q.2 B A, , 1. In figAB, , [….], NR, , BC and DC, , 𝑡ℎ𝑒𝑛, , B, , =, , ,, , 100, 110, , =, , [….], NR, , , NR = [.....] cm, , BC AB=6, DC=4, , A(∆ABC), =?, A(∆BCD), , C, D, , 2. In fig seg AC & seg BD intersect each other at point p, , AP, PC, , BP, , =PD then prove that ∆ABP~ ∆DPC, , 3. ∆ABP~ ∆DEF & A( ∆ABP): A(∆DEF) = 144: 81 then AB: DE =?, 4. From given information is PQ ⃦ BC ?, A, p, , P, , AP=2, PB=4 AQ=3,QC=6, Q
Page 7 :
B, 5., , C, , Areas of two similar triangles are 225 𝑐𝑚2 and ,81 𝑐𝑚2 if side of smaller, , triangle is 12cm. find corresponding side of major triangle, , 6., , D, , from adjoining figure, ⦟ABC = 90° ⦟DCB = 90° AB = 6,, , A, 6, , 8, , B, , DC=8, , then, , A(∆ABC), A(∆BCD), , =?, , C, , Q.3A) Complete the following activity 3 marks, ., 1. ∆ABC APpendicular BC & BQ perpendicular AC , B-P-C,A-Q-C, then, , show that ∆CPA~ ∆CQB if AP=7,BQ=8 BC=12, 0, then AC=? 𝐼𝑛∆CPAand ∆CQB ⦟CPA ≅ [⦟ … ].(each 90 ), ⦟ACP ≅ [⦟ … ].(common angle), ∆CPA~ ∆CQB (..........similarity test ), AP, BQ, , 7, 8, , =, =, , [….], BC, , (corresponding sides of similar triangle), , [….], 12, , ACx[.....]=7x12, , AC=10.5, , 2. A line is parallel to one side of triangle which intersects remaining two sides in two, distinct point then that line divdes sides in same proportion.
Page 8 :
Given :In ∆ABC line l II side BC & line l intersect side AB in P & side, AC in Q, P, , A, Q, , AP, , Q, , AQ, , Given: PB = QC, , construction :draw CP & BQ, , 𝑃𝑟𝑜𝑜𝑓: ∆APQ&∆PQB, , have equal, , height, , B, , C, , A(∆APQ), A( ∆PQB), , =, , A(∆APQ), A( ∆PQC), , =, , [….], PB, , [….], QC, , (areas in proportion of base)I, (areas in proportion of baseII, , ∆PQC&∆PQB have [.....]is common base, SegPQ II Seg BC hence height of:, , ∆APQ&∆PQB, 𝐴(∆PQC)=A( ∆......)...........(III), A(∆APQ), A( ∆PQB), AP, PB, , =, , A(∆ …….), A( ∆ ……… ), , ..............[(I),(II)&(III)], , AQ, , = QC ............[(I) & (II), , ,, From fig.seg PQ II side BC, AP= x +3 ,PB=x -3,AQ= x +5 ,QC=x-2, , 3., , then complete the activity to find the, , A, x+3, , value of x, , in∆PQB, PQ II side BC, , x+5, , AP, PB, , P, x-3, , AQ, , = [… ], , Q, , ..........([...........]), , x+3 x+5, =, x−3, [… ], , x-2, , (x+3)[......]=(x+5)(x-3), , B, , C, , 𝑥 2 +x-[....]=𝑥 2 +2x-15, x=[....]
Page 9 :
Q.3 B 3 marks, 1. There are two poles having heights 8m & 4m on, plane ground as shown in fig. Because of sunlight, shadow of smaller pole is 6m long then find the length, of shadow of longer pole., , 2.In ∆ABC B-D-C & BD=7, BC=20 then find the following ratio, 1), 2), 3), , A(∆ABD), A( ∆ADC), A(∆ABD), A( ∆ABC), A(∆ADC), A( ∆ABC), , 3. In given fig.quadrilateral PQRS side PQ II ⃦side SR ,AR=5 AP,, then prove that ,, , SR=5PQ, , 4., In triangle ABC point D is on side BC (B-D-C) such that, , ⦟BAC = ⦟ADC then prove that CA2 = CBxCD, , 5., , A, , D, , In Quadrlateral ABCD Side AD II BC diagonal AC &, AP, , BD intersct in point P then prove that PD =, , P, , B, , C, , PC, BP
Page 10 :
Q.4 4 marks, 1. Side of eqilateral triangle PQR is 8 cm then find the area of triangle whose side is, half of side of triangle PQR, 2.Areas of two similar triangle are equal then prove that triangles are congruent, 3.Two triangles are similar .Smaller triangle sides are 4 cm ,5 cm,6 cm perimter of, larger triangle is 90 cm then find the sides of larger triangle., Q.5 3 marks, 1. ln fig , PS = 2, SQ=6 QR = 5, PT = x & TR = y. then find the pair of value of x&y, such that ST ll side QR., P, , S, Q, , T, 5, , R, , 2 .An architecture have model of building, length of building is 1m then length of, model is 0.75cm then find length & height of model building whose actual length is, 22.5m& heght is 10m.
Page 11 :
2. PYTHAGORAS THEOREM, Que. 1 (A). Choose the correct alternative from those given below, (1 mark each ), 1. Out of given triplets, which is a Pythagoras triplet ?, (A) (1,5,10), , (B) (3,4,5), , (C) (2,2,2), , (D) (5,5,2), , 2. Out of given triplets, which is not a Pythagoras triplet ?, (A) (5,12,13), , (B) (8,15,17) (C) (7,8,15), , (D) (24,25,7), , 3. Out of given triplets, which is not a Pythagoras triplet ?, (A) (9,40,41) (B) (11,60,61), , (C) (6,14,15), , (D) (6,8,10), , 4. In right angled triangle, if sum of square of sides of right angle is, 169 then what is the length of hypotenuse?, (A) 15, , (B) 13, , (C) 5, , (D) 12, , 5. A rectangle having length of a side is 12 and length of diagonal is, 20 then what is length of other side?, (A)2, , (B) 13, , (C) 5, , (D) 16, , 6. If the length of diagonal of square is √2 then what is the length of, each side ?, (A)2, , (B)√3, , (C) 1, , (D) 4, , 7. If length of both diagonals of rhombus are 60 and 80 then what is, the length of side?
Page 12 :
(A)100, , (B)50, , (C) 200, , (D) 400, , 8. If length of sides of triangle are a ,b, c and a2 + b2 = c2 then which, type of triangle it is ?, (A)Obtuse angled triangle, , (B) Acute angled triangle, , (C) Equilateral triangle, , (D)Right angled triangle, , 9. In ∆ABC, AB = 6√3 cm, AC = 12 cm, and BC = 6 cm then mA, =?, (A)300, , (B) 600, , (C) 900, , (D) 450, , 10. The diagonal of a square is 10 √2 cm then its perimeter is ......... ., (A)10 cm., , (B) 40√2 cm., , (C) 20 cm., , (D) 40 cm., , 11. Out of all numbers from given dates, which is a Pythagoras triplet, ?, (A)15/8/17, , (B)16/8/16, , (C) 3/5/17, , (D) 4/9/15, , Que. 1 (B). Solve the following questions : (1 mark each ), 1.Height and base of a right angled triangle are 24 cm and 18 cm find, the length of its hypotenus ?, 2. From given figure, In ∆ ABC, AB⊥ BC, AB =BC then m A = ?, A, , B, , C, , 3. From given figure, In ∆ ABC, AB⊥ BC, AB =BC, AC = 2√2 then, Ɩ (AB) = ?, , A
Page 13 :
B, , C, , 4. From given figure, In ∆ ABC, AB⊥BC, AB =BC, AC = 5√2 then, what is the height of ∆ ABC ?, A, , B, , C, , 5. Find the height of an equilateral triangle having side 4 cm. ?, 6. From given figure, In ∆ ABQ, If AQ = 8 cm. then AB = ?, A, 300, B, , Q, , 7. In right angled triangle, if length of hypotenuse is 25 cm. and, height is 7 cm. then what is the length of its base ?, 8. If a triangle having sides 50 cm., 14 cm, and 48 cm., then state, wheather given triangle is right angled triangle or not., 9. If a triangle having sides 8 cm., 15 cm., and 17 cm., then state, wheather given triangle is right angled triangle or not., 10. A rectangle having dimensions 35 m X 12 m, then what is the, length of its diagonal ?, , Que. 2 (A). Complete the following activities ( 2 marks each ), * ( Write complete answers, don’t just fill the boxes ), 1. From given figure, In ∆ ABC, If AC = 12 cm. then AB = ?
Page 15 :
- BD2 = AC2-, , ∴, , ∴ AB2+CD2 =AC2+ BD2, 3. From given figure, In ∆ ABC, If ABC = 900 CAB=300 , AC =, 14 then for finding value of AB and BC, complete the following, activity., A, 30°, , B, , C, , Activity : In ∆ ABC, If ABC = 900 CAB=300, ∴ BCA =, By theorem of 300– 600– 900 ∆le,, 1, , ∴, , = AC, 2, , 1, , ∴ BC = ×, 2, , and, & AB =, , =, √3, 2, , √3, AC, 2, , × 14, , ∴ BC = 7 & AB =7 √3 ., 4. From given figure, In ∆ MNK, If MNK = 900 M=450 , MK = 6, then for finding value of MK and KN, complete the following, activity.
Page 16 :
M, 45°, , N, , K, , Activity : In ∆ MNK, If MNK = 900 M=450 ...( given ), ∴K=, , .... ( remaining angles of ∆ MNK ), , By theorem of 450– 450– 900 ∆le,, ∴, , =, ∴ MN =, , 1, √2, , 1, √2, , MK and, , ×, , & KN =, , =, 1, √2, , 1, √2, , MK, , × 6, , ∴ MN = 3 √2 . & KN = 3√2 ., 5. A ladder 10 m long reaches a window 8m above the ground. Find, the distance of the foot of the ladder from the base of wall. Complete, the given activity., Activity : as shown in fig. suppose, P, , Q, , R
Page 17 :
PR is the length of ladder = 10 m, At P – window, At Q – base of wall, At R – foot of ladder, ∴ PQ = 6 m, ∴ QR = ?, In ∆PQR , m PQR = 900, By Pythagoras Theorem,, ∴ PQ2 +, , = PR2…… (I), , Here, PR =10 , PQ =, From equation (I), 82 + QR2 =102, QR2 = 102- 82, QR2 = 100– 64, QR2 =, QR =6, ∴ The distance of foot of the ladder from the base of wall is 6 m., 6. From the given figure, In ∆ ABC, If AD⊥BC, C = 450, AC =, 8√2, BD = 5 then for finding value of AD and BC, complete the, following activity.
Page 18 :
A, , B, , C, D, , Activity : In ∆ ADC, If ADC = 900 C=450 ... ( given ), ∴ DAC =, , .... ( remaining angles of ∆ ADC ), , By theorem of 450– 450– 900 ∆le,, ∴, , =, ∴ AD =, , 1, √2, , 1, √2, , AC, , ×, , and, & DC =, , =, 1, √2, , 1, √2, , AC, , × 8√2, , ∴ AD = 8 & DC = 8, ∴BC =BD +DC = 5 + 8 = 13, 7. Complete the following activity to find the length of hypotenuse of, right angled triangle, if sides of right angle are 9 cm and 12 cm., Activity : In ∆PQR , m PQR = 900, P, , Q, , R
Page 20 :
∴ PQ2 =, ∴ PQ = √164, Here, ∆QPR ~∆QMP ~∆PMR, ∴ ∆QMP ~∆PMR, ∴, , PM, RM, , =, , QM, PM, , ∴PM2=RM X QM, ∴102= RM X 8, RM =, , 100, 8, , =, , And,, QR =QM + MR, +, , QR =, , 25, 2, , =, , 41, 2, , 9. Find the diagonal of a rectangle whose length is 16 cmand area is, 192sq.cm. Complete the following activity., Activity :, , T, , N, , L, , M, , As shown in fig., , LMNT is rectangle, , ∴ Area of rectangle = length X breadth, ∴ Area of rectangle =, ∴ 192 =, , X breadth, , X breadth, , ∴ Breadth = 12 cm., Also, TLM = 900 ..... ( each angle of rectangle is right angle ), In ∆TLM, By Pythagoras theorem
Page 21 :
∴ TM2 = TL2 +, ∴ TM2 = 122 +, ∴ TM2 = 144 +, ∴TM2 = 400, ∴ TM = 20, 10. In ∆ LMN, l = 5, m = 13, n = 12 then complete the activity to, show that wheather given traingle is right angled traingle or not., * ( l , m, n are opposite sides of L, M, N respectively ), Activity :, In ∆LMN मध्ये, l = 5, m = 13, n =, ∴ l2 =, , m2= 169 ;, , ;, , n2= 144., , ∴ l2 + n2 = 25 + 144 =, ∴, , + l 2= m2, , ∴By Converse of Pythagoras theorem, ∆LMN is right angled triangle., Que. 3 (B). Solve the following questions : (3 marks each ), 1. As shwon in figure, DFE = 900, FG⊥ED, If GD = 8, FG = 12,, then (1) EG = ? (2) FD = ? (3) EF = ?, G, , E, , D, , F, , 2. A congruent side of an isosceles right angled triangle is 7 cm ,Find, its perimetre .
Page 22 :
Que. 4. Solve the following questions : (Challenging question 4, marks each ), 1. As shwon in figure, LK = 6 √2 then 1) MK = ? 2) ML = ? 3) MN, =?, , N, 450, , M, , L, , K, 300
Page 23 :
3 Circle., Q.1. Four alternative answers for each of the following questions are given., Choose the correct alternative., 1) Two circles intersect each other such that each circle passes through the, centre of the other. If the distance between their centres is 12, what is the, radius of each circle ?, (A) 6 cm (B) 12 cm, (C) 24 cm, (D) can’t say, 2) A circle touches all sides of a parallelogram. So the parallelogram must be a,, ......... ., (A) rectangle (B) rhombus (C) square, (D) trapezium, 3) ∠ACB is inscribed in arc ACB of a circle with centre O. If ∠ ACB = 65°,, find m(arc ACB)., (A) 65°, (B) 130°, (C) 295°, (D) 230°, 4) In a cyclic ⃞ ABCD, twice the measure of ∠A is thrice the measure of ∠C., Find the measure of ∠C?, (A) 36, (B) 72, (C) 90, (D) 108, 5) How many circles can drawn passing through three non -collinear points?, (A) 0 (B) Infinite, (C) 2, (D) One and only, one(unique), 6) Two circles of radii 5.5 cm and 4.2 cm touch each other externally. Find the, distance between their centres, (A)9.7, (B) 1.3, (C) 2.6, (D) 4.6, 7) What is the measurement of angle inscribed in a semicircle?, (A) 90° (B) 120° (C) 100°, (D) 60°, 8) Two circles having diameters 8 cm and 6 cm touch each other internally., Find, the distance between their centres., (A), 2 (B) 14, (C) 7, (D) 1, 9) Points A, B, C are on a circle, such that m(arc AB) = m(arc BC) = 120°. No, point, except point B, is common to the arcs. Which is the type of ∆ ABC?, (A) Equilateral triangle, (B) Scalene triangle, (C) Right angled triangle, (D) Isosceles triangle, 10), , In, , PQRS if ∠RSP = 80° then find ∠RQT ?, (A), , 100°, , (C) 70°, , (B) 80°, (D) 110°
Page 24 :
Q.2 Solve the following sub-questions. (1 mark question), 1), 2), , How many circles can be drawn passing through a point?, Segment DP and segment DQ are tangent, segments to the circle with center A,, If DP = 7 cm. So find the length of the, segment DQ?, , 3) Two circles having radii 3.5 cm and 4.8 cm touch each other internally., Find the distance between their centres., 4) What is the measure of a semi circular arc?, 5), A, B, C are any points on the circle with centre, O. If m arc (BC) = 110° and m arc (AB) =, 125°, find measure arc AC, , 6), , In the figure if ∠PQR = 50° then find ∠PSR, , 7), In the adjoining figure the radius of a circle with, centre C is 6 cm, line AB is a tangent at A. What is, the measure of ∠ CAB? Why?, , 8), , In the figure quadrilateral ABCD is a cyclic , if, ∠DAB = 75° then find measure of ∠DCB
Page 25 :
9), , In the adjoining figure, seg DE is the chord of, the circle with center C. seg CF⊥ seg DE and, DE = 16 cm, then find the length of DF?, , 10), , In the figure, if ∠ABC = 35° then find, m(𝑎𝑟𝑐 AXC ) ?, , Q.3 Complete the following activities (2 marks each)., , The chords corresponding to congruent arcs of a circle are congruent.Prove the, theorem by completing following activity., Given : In a circle with centre B, , arc APC ≅ arc DQE, To Prove : Chord AC ≅chord DE, Proof : In ∆ ABC and ∆ DBE,, side AB ≅ side DB, side BC ≅ side, ∠ ABC ≅ ∠ DBE, , (measure of congruent arcs), , ∆ ABC ≅ ∆ DBE, , chord AC ≅ chord DE, 2), , In figure , points G, D, E, F, are concyclic points of a circle with centre C., ∠ ECF = 70°, m(arc DGF) = 200°, , find m(arc DEF) by completing activity., , (c.s.c.t)
Page 27 :
4), , In figure, chord EF || chord GH. Prove that,, chord EG≅ chord FH. Fill in the blanks and write the proof., Proof : Draw seg GF., , ∠𝐄𝐅𝐆 = ∠𝐅𝐆𝐇, , (I), , ......, , ∠𝐄𝐅𝐆 =, , .....( inscribed angle theorem ) (II), , ∠𝐅𝐆𝐇 =, , .....( inscribed angle theorem) (III), , ∴ m(arc EG) =, , .....[ By (I) , (II) व (III) ], , chord EG ≅ chord FH ......(corresponding chords of congruent arcs, , ), , 5), The angle inscribed in the semicircle is a right angle Prove the result by completing the, following activity ., Given: ∠ABC is inscribed angle in a, semicircle with center M., , To prove : ∠ABC is a right angle., Proof: segment AC is a diameter of the circle., ∴ m(arc AXC) =, Arc AXC is intercepted by the inscribed angle ∠ABC, ∠ABC, , =, =, , ∴ m ∠ABC, , ., , .....(Inscribed angle theorem), 𝟏, 𝟐, , ×, , =, , ∴ ∠ABC is a right angle.
Page 28 :
6), , Prove that angles inscribed in the same arc are congruent., Given: In a circle with centre C, ∠PQR, and ∠PSR, , is inscribed in same arc, , PQR.Arc PTR is intercepted by the angles., To prove : ∠PQR ≅ ∠PSR., , Proof :, , m∠PQR =, m∠, , 𝟏, 𝟐, , × [m(arc PTR)] ........ (i), =, , m∠, , 𝟏, 𝟐, , × [m(arc PTR)] ....... (ii), = m∠PSR, , ........By(i) &(ii), , ∴ ∠PQR ≅ ∠PSR, , 7), , If O is the center of the circle in the figure alongside , then complete the table from, the given information., The type of arc, , Type of circular arc, , Name of circular arc Measure of circular arc, , Minor arc, Major arc, Q.4. Solve the following sub-questions. (2 marks question), 1), In the adjoining figure circle with Centre D, touches the sides of ∠ACB at A and B. If, ∠ ACB = 52°, find measure of ∠ ADB.
Page 29 :
2), , In the adjoining figure, the line MN touches the, circle with center A at point M. If AN = 13 and, MN = 5 then find the radius of the circle?, , 3) What is the distance between two parallel tangents of a circle having radius, 4.5 cm? Justify your answer., 4), In figure, m(arc NS) = 125°, m(arc EF) = 37°,, find the measure ∠NMS., , 5) Length of a tangent segment drawn from a point which is at a distance 15 cm, from the centre of a circle is 12 cm, find the diameter of the circle?, 6), , In the figure a circle with center C has, m (arc AXB) = 100° then find central ∠ACB and, measure m (arc AYB)., , 7), , In figure , M is the centre of the circle and seg KL, is a tangent segment. If MK = 12, KL = 6√3 then, find (1) Radius of the circle., (2) Measures of ∠K and ∠M., , 8), , In figure, chords AC and DE intersect at B., If ∠ ABE = 108°, m(arc AE) = 95°, find m(arc DC) .
Page 30 :
Q. 5. Complete the following activity. (3 marks each), 1) Tangent segments drawn from an external point to a circle are congruent , prove, this theorem.Complete the following activity., Given :, To Prove:, Proof : Draw radius AP and radius AQ and complete the following proof of, the theorem., In ∆PAD and ∆QAD ,, Seg PA ≅, , .... ( radii of the same circle. ), , Seg AD ≅ Seg AD, , .... (, , ....(tangent theorem ), , ∠APD ≅ ∠AQD = 90°, ...., , ∴ ∆PAD ≅ ∆QAD, , ), , ∴seg DP ≅ seg DQ, , (, , ), , .... (, , ), , 2), MRPN is cyclic, ∠R = (5x - 13) °, ∠ N = (4x + 4) °. Find measures of ∠, R and ∠ N, by completing the following activity., Solution :, MRPN is cyclic, The opposite angles of a cyclic square are, ∠R +, , ∠N, , =, , ∴(5x-13)° + (4x+4°) =, ∴ 9x = 189, ∴ x, ∴ ∠R = (5x-13)° =, ∴ ∠N = (4x+4)°, , =, , =
Page 31 :
3), , In figure , seg AB is a diameter of, a circle with centre O . The bisector, of ∠ACB intersects the circle at point D., Prove that, seg AD ≅ seg BD., Complete the following proof by filling, in the blanks., , Proof Draw seg OD., ∠ACB =, , .......... angle inscribed in semicircle, , ∠DCB =, , m(arc DB) =, , .......... CD is the bisector of ∠C, .......... inscribed angle theorem, , ∠DOB =, , seg OA ≅ seg OB, , .......... definition of measure of an arc (I), .........., , ∴line OD is, , (II), , of seg AB .......... From (I) and (II), , ∴seg AD ≅ seg BD, , 4), , In the adjoining figure circles with centres X and Y, touch each other at point Z. A secant passing, through Z intersects the circles at points A and, B respectively., Prove that , radius XA || radius YB., Fill in the blanks and complete the proof., , Construction : Draw segments XZ and YZ., Proof :By theorem of touching circles, points X, Z, Y are, ∴ ∠XZA ≅, , .......... opposite angles, , Let ∠XZA = ∠BZY = a, , ..... (I), , Now, seg XA ≅ seg XZ, , ........ (radii of the same circle.), , ∴ ∠XAZ = .......... = a, , ........ (isosceles triangle theorem) (II), , similarly, seg YB ≅ seg YZ, , ........ (radii of the same circle.), , ∴ ∠BZY = .......... = a, , ........ (isosceles triangle theorem.) (III)
Page 32 :
∴from (I), (II), (III),, ∠XAZ =, ∴ radius XA || radius YB .......... (, , 5), , ), , An exterior angle of a cyclic quadrilateral is congruent to the angle opposite to, its adjacent interior angle, to prove the theorem complete the activity ., Given :, , ABCD is cyclic ,, is the exterior angle of ABCD, , To prove : ∠DCE ≅ ∠BAD, , Proof :, , + ∠BCD =, , ....(Angles in linear pair) (I), , ABCD is a cyclic ., + ∠BAD =, By (I) and (II), ∠DCE + ∠BCD =, , .....(Theorem of cyclic quadrilateral) (II), , + ∠BAD, , ∠DCE ≅ ∠BAD, , 6), , Seg RM and seg RN are tangent segments, of a circle with centre O. Prove that seg, OR bisects ∠MRN as well as ∠MON with, the help of activity.
Page 34 :
Q.6. Solve the following sub-questions. (3 marks question), 1) Prove the following theorems:, i) Opposite angles of a cyclic quadrilateral are supplementry., ii) Tangent segments drawn from an external point to a circle are congruent., iii) Angles inscribed in the same arc are congruent., 2), , 3), , Line ℓ touches a circle with centre O at point P. If, radius of the circle is 9 cm, answer the following., (i) What is d(O, P) = ? Why ?, (ii) If d(O, Q) = 8 cm, where does the point Q lie ?, (iii) If d(PQ) = 15 cm, How many locations of point R, are line on line ℓ? At what distance will each of, them be from point P ?, , In the adjoining figure, O is the centre of the, circle. From point R, seg RM and seg RN are, tangent segments touching the circle at M and, N. If (OR) = 10 cm and radius of the circle = 5, cm, then, (1) What is the length of each tangent segment ?, (2) What is the measure of ∠MRO ?, (3) What is the measure of ∠ MRN ?, , 4), In figure ,chord AB ≅ chord CD,, Prove that, arc AC ≅ arc BD
Page 35 :
5), In figure , in a circle with centre O, length of, chord AB is equal to the radius of the circle. Find, measure of each of the following., (1) ∠AOB, , (2) ∠ACB, , (3) arc AB, , 6), , In figure , chord LM ≅ chord LN , ∠L = 35°, find (i) m(arc MN), (ii) m(arc LN), , 7) Prove that, any rectangle is a cyclic quadrilateral., 8), , In figure , PQRS is cyclic., side PQ ≅ side RQ. ∠ PSR = 110°,, Find- (1) measure of ∠ PQR, (2) m(arc PQR), (3) m(arc QR), , 9), In figure , line ℓ touches the circle with, centre O at point P. Q is the mid point of, radius OP. RS is a chord through Q such that, chords RS || line ℓ. If RS = 12 find the radius, of the circle, 10), , In figure , O is the centre of a circle,, chord PQ ≅chord RS If ∠ POR = 70°, and (arc RS) = 80°, find (1) m(arc PR) (2), m(arc QS) (3) m(arc QSR)
Page 36 :
11), In the adjoining figure circle with Centre Q, touches the sides of ∠MPN at M and N. If, ∠ MPN = 40°, find measure of ∠ MQN., , 12), , In the figure if O is the center of the circle, and two chords of the circle EF and GH, are parallel to each other. Show that, ∠𝐄𝐎𝐆 ≅ ∠𝐅𝐎𝐇, , Q. 7. Solve the following sub-questions. (4 marks question), 1), In the figure segment PQ is the diameter of, the circle with center O. The tangent to the, tangent circle drawn from point C on it ,, intersects the tangents drawn from points P, and Q at points A and B respectively ,, prove that ∠AOC = 90°, , 2) The chords AB and CD of the circle intersect at point M in the interior of, the same circle then prove that CM × BD = BM × AC., 3), , A circle with centre P is inscribed in the, ∆ABC. Side AB, side BC and side AC touches, the circle at points L, M and N respectively., Radius of the circle is r., 1, , Prove that : A(∆ABC) = (𝐴𝐵 + 𝐵𝐶 + 𝐴𝐶) × r, 2
Page 37 :
4), In the figure ABCD is a cyclic, quadrilateral. If m(arc ABC) = 230°.then, find ∠ABC , ∠CDA , ∠CBE, , 5), The figure∆ABC is an isosceles triangle with a, perimeter of 44 cm. The sides AB and BC are, congruent and the length of the base AC is 12, cm. If a circle touches all three sides as shown, in the figure, then find the length of the tangent, segment drawn to the circle from the point B, , 6), , In the figure ∆ABC is an equilateral, triangle.The angle bisector of ∠𝐁 will, intersect the circumcircle ∆ABC at point P., Then prove that : CQ = CA., , 7), , In the figure quadrilateral ABCD is cyclic, , If m(arc BC) = 90° and ∠DBC = 55°., Then find the measure of ∠BCD .
Page 38 :
8), , Given : A circle inscribed in a right, angled ∆ABC. If ∠ACB = 90° and the, radius of the circle is r., To prove : 2 r = 𝒶 + b – c, , 9) In a circle with centre P , chord AB is parallel to a tangent and intersects the, radius drawn from the point of contact to its midpoint. If AB = 16√3 then, find the radius of the circle., 10), , In the figure, O is the center of the circle., Line AQ is a tangent. If OP = 3, m(arc PM) = 120°, then find the length of AP?, , Q. 8. Solve the following sub-questions (3 marks each), 1), , In the figure, O is the centre of the circle, and ∠AOB = 90° , ∠ABC = 30°, Then find ∠CAB?
Page 39 :
2), , 3), , 4), , In the figure a circle with center P, touches the semicircle at points Q, and C having center O. if diameter, AB = 10, AC = 6 then find the, radius 𝓍 of the smaller circle?, , In the figure a circle touches all the sides of, quadrilateral ABCD from the inside. The, center of the circle is O. If AD⊥ DC and, BC = 38 , QB = 27, DC = 25 then find the, radius of the circle?, , If AB and CD are the common, tangents in the circles of two unequal, (different) radii then show that, seg AB ≅ seg CD, , 5) Circles with centres A, B and C touch each other externally. If AB = 36,, BC = 32, CA = 30, then find the radii of each circle.
Page 40 :
4. Geometric Constructions, , Question 1) (A) choose the correct alternative answer for each of the following sub, question. Write the correct alphabet., 1) …………… number of tangents can be drawn to a circle from the point on the, circle., A), , 3, , B) 2, , C) 1, , D) 0, , 2) The tangents drawn at the end of a diameter of a circle are………….., A) Perpendicular, , 3), , ∆LMN ~∆HIJ, , and, , B) parallel, , 𝐿𝑀, 𝐻𝐼, , =, , 2, 3, , C) congruent D) can’t say, , then, , A), , ∆ LMN, , is a smaller triangle., , B), , ∆ HIJ, , is a smaller triangle., , C) Both triangles are congruent., D) Can’t say., 4) ……………….number of tangents can be drawn to a circle from the point, outside the circle., A) 2, , B) 1, , C) one and only one, , D) 0
Page 41 :
5), , In the figure ∆ ABC ~∆ ADE then the ratio of their corresponding sides is, --------., , 3, A), 1, 6), , Which, , 1, , B), theorem, , is, , C), , 3, used, , while, , constructing, , circle by using center of a circle?, A) tangent – radius theorem., B) Converse of tangent – radius theorem., C) Pythagoras theorem, D) Converse of Pythagoras theorem., 7) ∆PQR ~ ∆ABC,, , 𝑃𝑅, 𝐴𝐶, , =, , 5, 7, , then, , A) ∆ABC is greater., B), , ∆ PQR is greater., , C) Both triangles are congruent., , 3, , 4, D), 3, , 4, a, , tangent, , to, , the
Page 42 :
D) Can’t say., 8) ∆ABC ~∆AQR., A) A-Q-B, , 𝐴𝐵, 𝐴𝑄, , =, , 7, 5, , then which of the following option is true., , B) A-B-Q C) A-C-B D) A-R-B, , Question 1 (B) solve the following examples (1 mark each), 1) Construct ∠ABC =60 0 and bisect it., 2) Construct ∠PQR = 115 and divide it into two equal parts., 0, , 3) Draw Seg AB of lenght 9.7cm. Take point P on it such that AP =, 3.5 cm and A-P-B. Construct perpendicular to seg AB from point, P., 4) Draw seg AB of length 4.5 cm and draw its perpendicular bisector., 5) Draw seg AB of length 9 cm and divide it in the ratio 3:2., 6) Draw a circle of radius 3 cm and draw a tangent to the circle, from point P on the circle., , Question 2) (A) Solve the following examples as per the instructions given, in the activity. (2 marks each), 1), , Draw a circle and take any point P on the circle. Draw ray OP, , Draw perpendicular to ray OP from point P.
Page 43 :
2), , Draw a circle with center O and radius 3cm, , Take any point P on the circle., , Draw ray OP., , Draw perpendicular to ray OP from point P, , 1) To draw tangents to the circle from the end points of the, diameter of the circle., Construct a circle with center O. Draw any diameter AB of, it., , Draw ray OA and OB, , Construct perpendicular to ray OA from point A
Page 44 :
Construct perpendicular to Ray OB from point B, Question 2) (B) Solve the following examples (2 marks each), 1) Draw a circle of radius 3.4 cm take any point P on it. Draw tangent to the circle from point, P., 2) Draw a circle of radius 4.2 cm take any point M on it. Draw tangent to the circle from, point M., 3) Draw a circle of radius 3 cm. Take any point K on it. Draw a tangent to the circle from, point K without using center of the circle., 4) Draw a circle of radius 3.4 cm. Draw a chord MN 5.7 cm long in a circle. Draw a tangent to, the circle from point M and point N., 5) Draw a circle of 4.2 cm. Draw a tangent to the point P on the circle without using the, center of the circle., 6) Draw a circle with a diameter AB of length 6 cm. Draw a tangent to the circle from the, endpoints of the diameter., 7) Draw seg AB = 6.8 cm. Draw a circle with diameter AB. Draw points C on the circle apart, from A and B. Draw line AC and line CB Write the measure of angle ACB., , Question 3) (A) Do the activity as per the given instructions. (3 marks, each), 1) Complete the following activity to draw tangents to the circle., a) Draw a circle with radius 3.3 cm and center O. Draw chord PQ of length 6.6cm.., Draw ray OP and ray OQ., b) Draw a line perpendicular to the ray OP from P.
Page 45 :
c) Draw a line perpendicular to the ray OQ from Q., , 2) Draw a circle with center O. Draw an arc AB of 1000 measure., Perform the following steps to draw tangents to the circle from point A, and B., a) Draw a circle with any radius and center P., b) Take any point A on the circle., c) Draw ray PB such ∠ APB = 1000., d) Draw perpendicular to ray PA from point A., e) Draw perpendicular to ray PB from point B., , 3) Do the following activity to draw tangents to the circle without using, center of the circle., a) Draw a circle with radius 3.5 cm and take any point C on it., b) Draw chord CB and an inscribed angle CAB, c) With the center A and any convenient radius draw an arc intersecting, the sides of angle BAC in points M and N., d) Using the same radius draw and center C, draw an arc intersecting, the chord CB at point R., e) Taking the radius equal to d(MN) and center R, draw an arc, intersecting the arc drawn in the previous step. Let D be the point of, intersection of these arcs. Draw line CD. Line CD is the required, tangent to the circle., , Question 3 B) Solve the following examples (3 marks each):
Page 46 :
1) △ ABC ~ △ PBQ, In △ ABC, AB = 3 cm, ∠ B = 900, BC = 4 cm., Ratio of the corresponding sides of two triangles is 7:4. Then construct, △ ABC and △ PBQ, 2) ∆RHP ~∆NED,𝐼𝑛 ∆NED,NE=7 cm ,∠D=30 0 , ∠N=20 0 and, , = ., , 𝐻𝑃 4, 𝐸𝐷 5, , Then, , construct ∆RHP and ∆NED., 3) ∆PQR~∆ABC, In ∆PQR PQ=3.6cm, QR=4 cm, PR=4.2 cm ratio of, the corresponding sides of triangle is 3:4 then construct ∆PQR 𝑎𝑛𝑑 ∆ABC., 4) Construct an equilateral △ ABC with side 5cm. △ ABC ~ △ LMN, ratio, the corresponding sides of triangle is 6:7 then construct ∆LMN 𝑎𝑛𝑑 ∆ABC, 5) Draw a circle with center O and radius 3.4. Draw a chord MN of, length 5.7 cm in a circle. Draw a tangent to the circle from point M, and N., , 6) Draw a circle with center O and radius 3.6 cm. draw a tangent to, the circle from point B at a distance of 7.2 cm from the center of the, circle., , 7) Draw a circle with center C and radius 3.2 cm. Draw a tangent to, the circle from point P at a distance of 7.5 cm from the center of the, circle.
Page 47 :
8) Draw a circle with a radius of 3.5 cm. Take the point K anywhere, on the circle. Draw a tangent to the circle from K (without using the, center of the circle)., , 9), , Draw a circle of radius 4.2 cm. Draw arc PQ measuring 1200, Draw a tangent to the circle from point P and point Q., , 10) Draw a circle of radius 4.2 cm. Draw a tangent to the circle from, a point 7 cm away from the center of the circle., , 11) Draw a circle of radius 3 cm and draw chord XY 5 cm long. Draw, the tangent of the circle passing through point X and point Y (without, using the center of the circle)., , Question 4) solve the following examples. (4 marks each), 1), , ∆AMT ~∆AHE, In ∆AMT, AM =6.3 cm, , ∠MAT= 120 , AT = 4.9 cm,, 0, , =, , AM 7, HA 5, , then construct ∆AMT and ∆AHE ., , 2) ∆RHP~∆NED, In ∆NED, NE=7 cm. ∠D=30 , ∠N=20 ,, 0, , 0, , ∆RHP and △NED ., 3) ∆ABC. ~∆PBR, BC=8 cm, AC=10 cm , ∠B=90 0 ,, , =, , 𝐵𝐶 5, 𝐵𝑅 4, , then construct △ABC and ∆PBR, , =, , 𝐻𝑃 4, 𝐸𝐷 5, , then construct
Page 48 :
4) ∆AMT. ~∆AHE, 𝐼𝑛 ∆AMT AM=6.3 cm, ∠TAM=50 ,AT=5.6cm,, 0, , = ,, , 𝐴𝑀 7, 𝐴𝐻 5, , then, , construct △AMT and ∆AHE., 5) Draw a circle with radius 3.3cm. Draw a chord PQ of length 6.6cm ., Draw, , tangents to the circle at points P and Q. Write your observation, , about the tangents., 6), , Draw a circle with center O and radius 3 cm. Take the point P and the, point Q at a distance of 7 cm from the center of the circle on the, opposite side of the circle at the intersection passing through the center, of the circle, , Draw a tangent to the circle from the point P and the point, , Q., , Question 5) Solve the following examples (3 marks each), 1) Draw a circle with radius 4cm and construct two tangents to a circle, such that when those two tangents intersect each other outside the, circle they make an angle of 600 with each other., 2) AB = 6 cm, ∠BAQ = 500. Draw a circle passing through A and B so, that AQ is the tangent to the circle., , 3) Draw a circle with radius 3 cm. Construct a square such that each of its, side will touch the circle from outside., 4) Take points P and Q on the same side of line AB Draw a circle, passing through point P and point Q so that it touches line AB.
Page 49 :
5) Draw any circle with radius greater than 1.8 cm and less than 3 cm., Draw a chord AB 3.6 cm long in this circle. Tangent to the circle passing, through A and B without using the center of the circle, , 6) Draw a circle with center O and radius 3 cm. Take point P outside the, circle such that d (O, P) = 4.5 cm. Draw tangents to the circle from point P., , 7) Draw a circle with center O and radius 2.8 cm. Take point P in the exterior, of a circle such that tangents PA and PB drawn from point P make an angle, ∠APB of measure 70 0 ., 8) Point P is at a distance of 6 cm from line AB. Draw a circle of radius, 4cm passing through point P so that line AB is the tangent to the circle., ………………………………………………………………………………………………………………………………………………, …
Page 50 :
Coordinate Geometry, Q. 1 A) MCQ, 1), , Point P is midpoint of segment AB where A(- 4,2) and B(6,2) then the, coordinates of P are ---------, , A) ( -1, 2 ), , B) ( 1, 2 ), , C) (1, - 2), , D) ( -1, - 2), , 2) The distance between Point P ( 2 , 2 ) and Q ( 5, x ) is 5 cm then the, value of x = ---------A) 2, , B) 6, , C) 3, , D) 1, , 3) The distance between points P ( -1 , 1 ) and Q(5, -7 ) is ------------.., A) 11 cm, , B) 10 cm, , C) 5 cm, , D) 7 cm, , 4) If the length of the segment joining point L (x , 7 ) and point, M( 1, 15 ) is 10 cm then the value of x is --------A) 7, , B) 7 or -5, , C) - 1, , D) 1, , 5) Find distance between point A ( -3 , 4 ) and origin O., A) 7 cm, , B) 10 cm, , C) 5 cm, , D) -5 cm, , 6) If point P ( 1 , 1 ) divide segment joining point A and point B ( -1 , -1 ), in, , the ratio 5 : 2 then the coordinates of A are --------A)( 3 ,3 ), , B)( 6, 6 ), , C)(2, 2 ), , D)(1, 1 ), , 7) If segment AB is parallel Y-axis and coordinates of A are (1, 3) then, the coordinates of B are ------------A)( 3 ,1 ), , B)( 5, 3), , C)(3, 0), , D)(1, -3), , 8) If point P is midpoint of segment joining point A (-4, 2) and point, B(6, 2 ) then the coordinates of P are -----------A)( -1, 2 ), , B)( 1 , 2 ), , C)(1 , -2), , D) (-1, - 2)
Page 51 :
9) If point P divides segment AB in the ratio 1:3 where A(-5 , 3) and, B(3 , -5) then the coordinates of P are ----------------A)( -2, -2 ), , B)( -1 , -1 ), , C) (-3 , 1 ), , D) ( 1, - 3 ), , 10) If the sum of x-coordinates of the vertices of a triangle is 12 and the, sum of Y-coordinates is 9 then the coordinates of centroid are ---------A)(12 , 9), , B)(9 , 12), , C)(4, 3 ), , D)(3 ,4 ), , Q. 1 B. Solve the following (1 mark each), 1) Find the coordinates of the point of intersection of the graph of the, equation X = 2 and y = -3., 2) Find distance between point A (7, 5) and B (2, 5)., 3) The coordinates of diameter AB of a circle are A (2, 7) and, B (4 , 5), , then find the coordinates of the centre., , 4) Write the X-coordinate and Y-coordinate of point P(- 5 , 4)., 5) What are the coordinates of origin?, 6) Find distance of point A( 6, 8 ) from origin:, 7) Find coordinates of midpoint joining ( -2 ,6 ) and ( 8 ,2 ), 8) Find the coordinates of centroid of a triangle whose vertices are, (4, 7) , (8, 4) and (7 ,11)., 9) Find distance between point O(0, 0) and B (-5 , 12)., 10) Find coordinates of midpoint of point (0, 2) and (12, 14)., Q. 2 A) Complete the activity (each of 2 mark), 1) Find distance between point Q (3 , - 7) and point R ( 3, 3), Solution: Suppose Q (x1 , y1 ) and point R ( x2 , y2 ), X1 = 3 , y1 = -7, Using distance formula,, d ( Q, R ) = √, , and x2 = 3 , y2 = 3
Page 53 :
4) The coordinates of the vertices of a triangle ABC are A (-7, 6) ,B(2, , -2) and C( 8 , 5) find coordinates of its centroid., Solution : - Suppose A( x1 , y1 ) and B( x2,y2) and C ( x3, y3 ), X1 = -7, y1 = 6 and x2 = 2 , y2 = -2 and x3 = 8 , y3 = 5, Using Centroid formula, ∴ 𝐶𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑒𝑛𝑡𝑟𝑜𝑖𝑑 𝑜𝑓 𝑎 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒, ABC= (, , 𝑥1+𝑥2+𝑥3, 3, , ,, , 𝑦1+𝑦2+𝑦3, 3, , ), , = (, , 3, , ,, , ), , 3, , ∴ 𝐶𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑒𝑛𝑡𝑟𝑜𝑖𝑑 𝑜𝑓 𝑎 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 ABC = (, , 3, 3, , ,, , ∴ 𝐶𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑒𝑛𝑡𝑟𝑜𝑖𝑑 𝑜𝑓 𝑎 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 ABC= ( 1,, , ), ), , Q. 2 Solve (Each of 2 marks), 1) The point Q divides segment joining A (3, 5) and B (7, 9) in the ratio, 2 : 3. Find the X-coordinate of Q., 2) If the distance between point L ( x , 7) and point M ( 1, 15 ) is 10, then find the value of X., 3) Find the coordinates of midpoint of segment joining (22, 20) and (0, ,16), 4) Find distance CD where C(-3a , a), D(a, -2a)., 5) Show that the point(11, -2) is equidistant from (4, -3) and (6, 3)., Q. 3 A) Complete the activity ( Each of 3 marks), 1) If the point P (6,7) divides the segment joining A (8, 9) and B(1, 2) in, some ratio. Find that ratio., Solution : Point P divides segment AB in the ratio m : n., A ( 8, 9 ) = ( x1 , y1 ) , B (1, 2 ) = ( x2 , y2 ) and P (6, 7 ) = (x,, y), Using Section formula of internal division,
Page 54 :
) +, , 𝑚(, , ∴ 7 =, , 𝑚+𝑛, , ∴7m +7n =, ∴7m -, , + 9n, = 9n -, , ∴, ∴, , 𝑛( 9), , = 2n, , 𝑚, 𝑛 =, , A ( -1 , 1 ), , 1) From the figure given alongside find the length of, the median AD of triangle ABC ., Complete the activity., Solution :- Here A (-1 , 1),B(5, -3), C (3, 5) and, , B ( 5 , -3 ), , D, , suppose D (x,y) are coordinates of point D., Using midpoint formula,, X=, , 5+3, , −3+5, , y=, , 2, , ∴ 𝑥 =, , 2, , ∴ 𝑦=, , Using distance formula,, ∴ 𝐴𝐷 =, , √(4 −, , ∴ 𝐴𝐷 =, , )2, , ( 1 − 1 )2, , ) 2 + ( 0 )2, , √ (, ∴ 𝐴𝐷 =, , +, , √, , ∴ 𝑇ℎ𝑒 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑚𝑒𝑑𝑖𝑎𝑛 𝐴𝐷 =, Q. 3 B) Solve the following (Each of 3 marks), 1) Show that P(-2 , 2 ), Q (2, 2) and R ( 2, 7) are vertices of a right, angled triangle., 2) Show that the point (0 , 9) is equidistant from the points (-4,1) and, (4 , 1)., , C(3 ,5 )
Page 55 :
3) Point P(-4 , 6) divides point A (-6 , 10) and B (m , n) in the ratio, 2:1 then find the coordinates of point B., , Q. 4 Solve (Each of 4 marks), 1) Show that points A(-4 , -7), B(-1,2 ), C (8, 5) and D (5 , -4) are the, vertices of a parallelogram ABCD., 2) Show that the points (0 , -1), (8, 3) , (6 , 7) and (-2 , 3) are vertices of a, rectangle., 3) Show that the points (2, 0 ), (-2 , 0) and (0, 2 ) are vertices of a triangle., State the type of triangle with reason., 4) If A(5,4 ) , B (-3 , -2) and C (1 -8) are the vertices of a ∆ 𝐴𝐵𝐶 . Segment, AD is median. Find the length of seg AD:, 5) Show that A (1, 2) , (1, 6) , C (1 + 2√3 , 4 ) are vertices of an equilateral, triangle., Q.5) Solve (Each of 3 marks), A (0,2), , 1), Seg OA is the radius of a circle with centre O., O, , The coordinates of point A is (0 , 2) then, decide whether the point B(1, 2) is on the circle?, , 2) Find the ratio in which Y-axis divides the point A( 3, 5) and point B(6 , 7). Find the coordinates of that point., 3) The points (7, -6) , (2, K) and (h,18) are the vertices of triangle. If, (1,5) are the coordinates of centroid. Find the value of h and k.., 4) Using distance formula decide whether the points (4, 3), (5, 1) and, (1, 9) are collinear or not ?
Page 58 :
6. If cot ( 90 – A ) = 1 then A = ?, 1, , 7. If 1 − cos2 θ = then θ = ?, 4, , 8. Prove that, , cos ( 90 – A ), , =, , sin A, , 9. If tan θ X, , sin ( 90 – A ), cos A, , ., , = sin θ then, , =?, , 10. (sec θ + tan θ) . (sec θ - tan θ) = ?, 11., , sin 750, cos 150, , =?, , Que.) 2 A). Complete the following activities ( 2 marks each ), * ( Write complete answers, don’t just fill the boxes ), 1. Prove that cos2 θ . (1 + tan2 θ ) = 1. Complete the activity given, below., L . H . S. =, , Activity, , = cos2 θ X, , ...(1 + tan2 θ =, , ), = (cos θ X, , )2, , = 12, =1, = R .H .S., , 2., , 5, sin2 θ, , − 5 cot 2 θ,, , Activity, , Complete the activity given below., 5, , sin2 θ, , − 5 cot 2 θ
Page 63 :
cot A, , 9. Prove that, , 1−cot A, , 10. Prove that √, , +, , 1+cos A, 1−cos A, , tan A, 1−tan A, , =−1., , = cosec A + cot A ., , 11. Prove that sin4 A − cos4 A = 1 − 2cos2 A ., 12. Prove that sec 2 θ − cos2 θ = tan2 θ + sin2 θ ., 13. Prove that cosec θ – cot θ =, 14. In ∆ ABC, cos C =, 1+ sec A, , 15. Prove that, 16. If sin A =, , sec A, , 3, 5, , 17. Prove that, , 12, 13, , =, , sin θ, , ., , 1+ cos θ, , and BC = 24 then AC = ?, sin2 A, , 1−cos A, , ., , then show that 4 tan A + 3 tan A = 6 cos A, 1+sin B, cos B, , +, , cos B, , = 2 sec B ., , 1 +sin B, , Que. 4 Solve the following questions : (Challenging questions, 4, marks each ), , 1. Prove that, sin2 A . tan A + cos2 A . cot A + 2 sin A . cos A = tan A + cot A, ., 2, , 2, , 2. Prove that sec A − cosec A =, 3. Prove that, , cot A + cosec A − 1, cot A− cosec A + 1, , =, , 2sin2 A−1, sin2 A . cos2 A, , 1+cos A, sin A, , ., , ., , 4. Prove that sin θ ( 1 – tan θ ) − cos θ ( 1 − cot θ ) = cosec θ − sec θ, ., , 5. If cos A =, , 2 √𝑚, 𝑚+1, , then Prove that cosec A =, , 𝑚+1, 𝑚−1, , .
Page 64 :
6. If sec A = 𝑥 +, , 1, 4𝑥, , then show that sec A + tan A = 2𝑥 or, , 1, 2𝑥, , ., , 7. In ∆ ABC , √2 AC = BC, sin A = 1, sin2 A + sin2 B + sin2 C = 2, then A = ? B = ? C= ?, 8. Prove that sin6 A + cos6 A = 1 – 3 sin2 A . cos2 A ., 9. Prove that, 10. Prove that, , 2 (sin6 A + cos6 A) – 3 (sin4 A + cos4 A ) + 1 = 0 ., cot A, 1−tan A, , +, , tan A, 1−co t A, , = 1+ tanA + cotA = secA . cosecA, , +1, Que. 5 Solve the following questions : (Creative questions, 3, marks each ), 1. If 3 sin A + 5 cos A = 5 then show that 5 sin A – 3 cos A = ±3., 2. If cos A + cos 2 A = 1 then sin2 A + sin4 A = ?, 3. If cosec A – sin A = p, 2, , 2, 3, , आणि sec, , A – cos A = q then prove that, , 2, , (𝑝𝑞 2 )3, , (𝑝 𝑞) +, =1, 4. Show that tan 70 X tan 230 X tan 600 X tan 670 X tan 830 = √3., 5. If sin θ + cos θ = √3 then show that tan θ + cot θ = 1 ., 6. If, , tan θ - sin2 θ = cos2 θ then show that sin2 θ =, , 1, 2, , ., , 7. Prove that, ( 1 − cos2 A ) . sec 2 B + tan2 B ( 1− sin2 A ) = sin2 A + tan2 B