Tiêu đề 02-Similar Triangles (Part-2)(1).pdf ✅

EXERCISE # 1 A. Very Short Answer Type Questions Q.l In a  ABC, D and E are points on the sides AB and AC respectively such that DE || BC. (i) If AD = 6cm, DB = 9cm and AE = 8 cm, find AC. (ii) If DB AD = 4 3 and AC = 15 cm find AE. (iii) If 3 2 DB AD = and AC = 18 cm, find AE (iv) If AD = 4 cm, AE = 8 cm, DB = x – 4 and EC = 3x – 19, find x. (v) If AD = 8 cm, AB = 12 cm and AE = 12 cm, find CE. (vi) If AD = 4 cm, DB = 4.5 cm and AE = 8 cm, find AC. (vii) If AD = 2 cm, AB = 6 cm and AC = 9 cm, find AE. (viii) If 5 4 BD AD = and EC = 2.5 cm, find AE. (ix) If AD = x, DB = x – 2, AE = x + 2 and EC = x – 1, find the value of x. (x) If AD = 8x – 7, DB = 5x – 3, AE = 4x – 3 and EC = (3x – 1), find the value of x. (xi) AD = 4x – 3, AE = 8x – 7, BD = 3x – 1 and CE = 5x – 3, find the value of x. Q.2 In a ABC, D and E are points on the sides AB and AC respectively. For each of the following cases show that DE || BC : (i) AB = 12 cm, AD = 8 cm, AE = 12 cm and AC = 18 cm. (ii) AB = 5.6 cm, AD = 1.4 cm, AC = 7.2 cm and AE = 1.8 cm (iii) AB = 10.8 cm, BD = 4.5 cm, AC = 4.8 cm and AE = 2.8 cm. (iv) AD = 5.7 cm, BD = 9.5 cm, AE = 3.3 cm and EC = 5.5 cm Q.3 In a  ABC, AD is the bisector of A, meeting side BC at D. (i) If BD = 2.5 cm, AB = 5 cm and AC = 4.2 cm, find DC. (ii) If BD = 2cm, AB = 5 cm and DC = 3 cm, find AC (iii) If AB = 3.5 cm, AC = 4.2 cm and DC = 2.8 cm, find BD. (iv) If BC = 10 cm, AC = 14 cm and BC = 6 cm, find BD and DC. (v) If AC = 4.2 cm, DC = 6 cm, BC = 10 cm, find AB. (vi) If AB = 5.6 cm, AC = 6 cm and DC = 3 cm, find BC. (vii) If AB = 5.6 cm, BC = 6 cm and BD = 3.2 cm find AC. (viii) If AB = 10 cm, AC = 6 cm and BC = 12 cm, find BD and DC. Q.4 In  ABC, B = 2 C and the bisector of B intersects AC and D. Prove that BA BC DA BD = . Q.5 In fig. if AB ⊥ BC and DE ⊥ AC. Prove that  ABC ~  AED. B C D E A Q.6 In fig. if P = RTS, prove that RPQ ~ RTS. R T P Q S Q.7 In fig. AD and CE are two altitudes of ABC. C D F E A B Prove that (i)  AEF ~  CDF (ii)  ABD ~  CBE (iii)  AEF ~  ADB
(iv)  FDC ~  BEC Q.8 In fig. if BD ⊥ AC and CE ⊥ AB, F B E A D C Prove that (i) AEC ~  ADB (ii) DB CE AB CA = Q.9 E is a point on side AD produced of a parallelogram ABCD and BE intersects CD at F. Prove that  ABE ~  CFB. Q.10 In fig. E is a point on side CB produced of an isosceles triangle ABC with AB = AC. If AD ⊥ BC and EF ⊥ AC, prove that  ABD ~  ECF. A F E B D C B. Short Answer Type Questions Q.11 In fig, AE is the bisector of the exterior CAD meeting BC produced in E. If AB = 10 cm, AC = 6 cm and BC = 12 cm, find CE. 6 cm x cm E 12 cm C B A D Q.12 D, E and F are the points on sides BC, CA and AB respectively of ABC such that AD bisects A, BE bisects B and CF bisects C. If AB = 5 cm, BC = 8 cm and CA = 4 cm, determine AF, CE and BD. Q.13 (i) In fig.1, if AB || CD, find the value of x. (ii) In fig.2, if AB || CD, find the value of x. A B D C Fig.1 D C A B O Fig.2 (iii) In fig.3, AB || CD. If OA = 3x – 19, OB = x – 4, OC = x – 3 and OD = 4, find x. D C A B O Fig.3 Q.14 In a ABC, D and E are points on sides AB and AC respectively such that BD = CE. If B = C, show that DE || BC. Q.15 In fig. if DC AD = EC BE and CDE = CED, prove that  CAB is isosceles. C D E A B Q.16 In  ABC, D is the mid-point of BC and ED is the bisector of the ADB and EF is drawn
parallel to BC cutting AC in F. Prove that EDF is a right angle. Q.17 The bisectors of the angles B and C of a triangle ABC, meet the opposite side in D and E respectively. If DE || BC, prove that the triangle is isosceles. Q.18 In fig. if PR QT = QS QR and 1 = 2. Prove that PQS ~ TQR T P R S Q 1 2 Q.19 If CD and GH (D and H lie on AB and FE) are respectively bisectors of ACB and EGF and  ABC ~ FEG, prove that (i)  DCA ~  HGF (ii) GH CD = FG AC (iii)  DCB ~  HGE Q.20 If  ABC, if AD ⊥ BC and AD2 = BD × DC, prove that BAC = 90o. Q.21 In fig. if AD ⊥ BC and DA BD = DC DA , prove that  ABC is a right triangle. Q.22 ABC is an isosceles right triangle, right angled at C. Prove that AB2 = 2 AC2 . Q.23 In an isosceles triangle ABC, with AB = AC, BD is perpendicular from B to the side AC. Prove that BD2 – CD2 = 2 CD · AD Q.24 In a ABC, the angles at B and C are acute. If BE and CF be drawn perpendiculars on AC and AB respectively, prove that A B C F E (i) BC2 = AB × BF + AC × CE. (ii) AC2 = AB2 + BC2 – 2AB. BF (iii) AB2 = BC2 + AC2 – 2AC . CF Q.25 ABC is a right triangle, right angled at C and AC = 3 BC. prove that ABC = 60o. Q.26 In a right-angled triangle if a perpendicular is drawn from the right angle to the hypotenuse, prove that the square of the perpendicular is equal to the area of rectangle contained by the two segments of the hypotenuse. C. Long Answer Type Questions C. Q.27 ABCD is a quadrilateral; P, Q, R and S are the points of trisection of sides AB, BC, CD and DA respectively and are adjacent to A and C; prove that PQRS is a parallelogram. Q.28 In  ABC, the bisector of B meets AC at D. A line PQ || AC meets AB, BC and BD at P, Q and R respectively. Show that (i) PR · BQ = QR·BP (ii) AB × CQ = BC × AP. Q.29 In fig. CD and GH are respectively the medians of ABC and EFG. If ABC ~ FEG. Prove that (i)  ADC ~  FHG (ii) GH CD = FE AB
C A D B F H E G (iii)  CDB ~  GHE Q.30 In trapezium ABCD, AB || DC and DC = 2 AB. EF drawn parallel to AB cuts AD in F and BC in E such that EC BE = 4 3 . Diagonal DB intersects EF at G. Prove that 7 FE = 10 AB. Q.31 Through the vertex D of a parallelogram ABCD, a line is drawn to intersect the sides BA and BC produced at E and F respectively. Prove that AE DA = BE FB = CD FC Q.32 In fig. ABC is a right triangle right angled at B and D is the foot of the perpendicular drawn from B on AC. If DM ⊥ BC and DN ⊥ AB. A N D B M C 2 3 1 prove that (i) DM2 = DN × MC (ii) DN2 = DM × AN Q.33 In fig. AD and BE are respectively perpendiculars to BC and AC. C E D A B Show that (i)  ADC ~  BEC (ii) CA × CE = CB × CD (iii)  ABC ~  DEC (iv) CD × AB = CA × DE Q.34 ABC is an isosceles triangle with AB = AC and D is a point on AC such that BC2 = AC × CD. Prove that BD = BC. Q.35 In  PQR, QM ⊥ PR and PR2 – PQ2 = QR2 . Prove that QM2 = PM × MR. Q.36 Prove that the sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of its sides. ANSWER KEY