6 CHAPTER. Triangles. A plane figure bounded by three line segments is called a triangle.


 Ernest Brooks
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1 6 CHAPTER We are Starting from a Point but want to Make it a Circle of Infinite Radius A plane figure bounded by three line segments is called a triangle We denote a triangle by the symbol In fig ABC has (i) three vertices namely, A,B and C (ii) three sides namely, AB, BC, CA (iii) three angles namely, A, B and C Types of on the basis of sides (i) Equilateral triangle A triangle whose all the three sides are equal is called equilateral In the figure ABC is an equilateral triangle in which AB = BC = CA (ii) Isosceles triangle A triangle having two sides equal is called an isosceles triangle In the figure, ABC is an isosceles triangle in which AB = AC (iii) Scalene triangle A triangle whose sides are of different lengths In the figure ABC is a triangle in which AB BC CA Types of on the Basis of Angles (i) Obtuseangled triangle A triangle in which one angle is an obtuse angle, is called an obtuse angled triangle In figure, ABC is a triangle in which B 9 (ii) Acute angle triangle a triangle in which all angles are less than 9 in measures is called acute angled triangle /
2 (iii) A right angled triangle : A triangle in which one angle is of exact 9 is called right angle triangle Some Other Important Terms of Median A median of a triangle is the line segment joining the midpoint of side with the opposite vertex (b) Centroid The point of intersection of all the three medians of a triangle is called its centroid Characteristics of Centroid (i) Centroid is the point at which the medians of triangle meet (ii) The medians of a triangle are concurrent (iii) The centroid divides the medians in the ratio 2 : 1 (iv) The median of an equilateral triangle are equal (v) The medians of an equilateral triangle coincide with the altitudes Altitudes The altitude of a triangle is the perpendicular drawn from a vertex to the opposite side (d) Orthocentre The point of intersection of all the three altitudes of a triangle is called its orthocenter Characteristics of Orthocentre (i) Orthocentre is the point at which the altitudes of a triangle meet (ii) The altitudes of a triangle are concurrent (iii) Orthocentre of an acute triangle lies in the interior of the triangle (iv) Orthocentre of an obtuse triangle lies in the exterior of the triangle Orthocentre of a right triangle lies on the vertex of the right angle (e) (f) Angle bisectot: the angle bisector of an angle of a triangle is a line that divided the angle in two equal part Incentre of a triangle The point of intersection of the bisectors the internal angles of a triangle in called its incentre Characteristics of Incentre (i) The point at which the three angle bisectors of a triangle intersect is called the incentre /
3 (ii) The triangle may be acute, obtuse, or right, the angle bisectors of a triangle must meet at a point lying inside the triangle (iii) The incentre of a triangle lies in the interior of the triangle (iv) The bisectors of the angles of a triangle are concurrent (v) From the incentre we can draw a on opposite sides (vi) We can call this perpendicular as inradius (g) Perpendicular bisector: the perpendicular bisector of a triangle is perpendicular drawn from the opposite vertex and divide the opposite side in two equal parts (h) Circumcentre of a triangle The point of intersection of the perpendicular bisectors of the sides of a triangle is called its circumcentre Characteristics of Circumcentre (i) The point at which the perpendicular bisectors of the sides of a triangle meet is called the cicumkcentre of the triangle (ii) The right bisectors of the sides of a triangle are concurrent (iii) With circumcentre as centre, we can drawn a circle passing through the vertices of a triangle (iv) The distance from centre to the vertices is called the circumradius The circle thus drawn with circumentre as centre and circumradius as radius is called circumcircle (i) Perimeter The sum of lengths of the sides of a figure is its perimeter Perimeter of ABC = AB + BC + CA Asseingment1 1 Prove that the sum of the angles of a triangle is 18 2 In ABC, B 75, C 32 find A 3 In the figure, show that A B C D E F /
4 4 In the figure, prove that a b c 36 5 The angles of a triangle are in the ratio 3 : 5 : 1 Find the measures of each angle of the triangle 6 The sum and difference of two angles of a triangle are 128 and 22 respectively Find all the angles of the triangle 7 If the bisector of the angle B and C of a ABC meet at a point O, then prove that BOC = A In ABC, B > C, if AM is the bisector of BAC and AN BC, prove that MAN = ( 2 B C ) 9 Fill in the blanks The sum of three angles of a triangle is (b) If two angles of a triangle are 51 and 38, the third angle is equal to If the angles of a triangle in the ratio 2 : 2 : 5, then the angles are (d) The angles of a triangle are 3x 5, 2x + 55 and 5x 5 degrees then x is equal to (e) A triangle cannot have more than right angles (f) A triangle cannot have more than obtuse angle 1 Which of the following statements are true or false? An exterior angle of a triangle is less than either of its interior opposite angles (b) Sum of the three angles of a triangle is 18 A triangle can have two right angles (d) A triangle can have two acute angles (e) A triangle can have two obtuse angles (f) An exterior angle of a triangle is equal to the sum of the two interior opposite angles Exterior angle of a Triangle: Definition If the side BC of a triangle ABC is produced to ray BD, then ACD is called an exterior angle of triangle ABC at C, and is denoted by exterior ACD A and B are called remote interior angles or interior opposite angles Note : At each vertex there are two exterior angles Theorem Given A To prove Proof If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles ABC whose side BC has been produced to D forming exterior angle ACD A B ACD In ABC B ACB 18 A (1) ACB ACD 18 (2) (straight angle) From (1) and (2), we get ACB ACD A B ACB ACD A B /
5 A B ACD IX ACADEMIC QUESTIONS Subjective Assengment2 1 In the figure, find BED 2 In figure, prove that x A B C 3 In the figure, find x and y, if AB DF and AD FG 4 In the figure, prove that DE BF 5 In the figure, AB DG, AC DE, EDH = 25 and BAC = 2, Find x and y 6 In the figure, CD AB, ABE = 13 and BAC = 7 Find x and y /
6 7 If ABD = 125 and ACE = 13, then BAC = Congruent Figures The geometric figures are said to be congruent if they are exactly of the same shape and size Congruent segments Two segments are congruent if they are of the same length and conversely Hence in Fig AB CD A B C D (b) Congruent angles Two angles are congruent if they have equal measures and conversely Hence in fig BAC FDE Congruent circles Two circles are congruent if they have equal radii and conversely Hence in fig If r 1 = r 2 Then c 1 circle c 2 circle Rules for Congruent triangles Rules 1 (SAS) When two sides and the included angle are given In ABC and DEF If AB = DE, A D, AC = DF Then ABC DEF /
7 Rule 2 (SSS) When three sides are given In ABC and DEF If AB = DE, BC = EF, CA = FD Then ABC DEF Rule 3 (ASA) When two angles and the included side is given In ABC and DEF If B E, BC = EF, C F Then ABC DEF Rule 4 (RHS) When Right Angle Hypotenuse Side are given In ABC and DEF If B E = 9, BC = EF, CA = FD Then ABC DEF Congruence Relations of (i) Reflexive ABC ABC (congruence relation is reflexive) Every triangle is congruent to itself (ii) Commutative If ABC DEF, then DEF ABC (congruence relation is commutative) (iii) Transitive If ABC DEF, and DEF PAR then ABC PAR (congruence relation is transitive) Note : Since the congruence relation is reflexive, commutative and transitive, it is an equivalence relation /
8 IX ACADEMIC QUESTIONS Subjective Asseingment3 1 Prove that ABC is isosceles if altitude AD bisects BC 2 Prove that ABC is isosceles if median AD is perpendicular to BC 3 In the fig it is given that AB = CF, EF = BD and AFE = DBC Prove that AFE CBD 4 ABCD is a quadrilateral in which AD = BC and DAB CBA Prove that : (i) ABD BAC (ii) BD = AC (iii) ABD BAC 5 In the fig AC = AE and AB = AD and BAD = EAC Prove that BC = DE 6 In fig l m and M is the mid point of the line segment AB Prove that M is also the midpoint of nay line segment CD having its end points on l and m respectively /
9 7 In fig It is given that BC = CE and 1 2 Prove that GCB DCE 8 In fig AD and BC are perpendicular to the line segment AB and AD = BC Prove that O is the mid point of line segment AB and DC 9 In the figure C is the mid point of AB BAD CBE ECA DCB Prove that (i) DAC EBC (ii) DA = EB 1 In the fig BM and DN are both perpendiculars to the segments AC and BM = DN Prove that AC bisects BD 11 In fig, PS = PR, TPS QPR Prove that PT = PQ 12 In fig AD = AE and D and E are points on BC such that BD = EC Prove that AB = AC 13 In the fig AD CD and BC CD If AQ = BP and DP = CQ Prove that DAQ CBP /
10 14 In the fig AB = AC, ABD ACD Prove that BD = CD 15 In fig QPR PQR and M and N are respectively on sides QR and PR of PQR such that QM = PN Prove that OP = OQ, where O is the point of intersection of PM and QN 16 In the fig AB = AC BE and CF are respectively the bisectors of B and C Prove that EBC FCB 17 AD and BE are respectively altitude of ABC such that AE = BD Prove that Ad = BE 18 AD, BE and CF, the altitudes of ABC are equal Prove that ABC is an equilateral triangle 19 AD is the bisector of A of a triangle ABC, P is any point on AD Prove that the perpendicular drawn from P on AB and AC are equal 2 ABCD is a parallelogram, if the two diagonals are equal, find the measure of ABC 21 Fill in the blanks in the following so that each of the following statements is true (i) (ii) (iii) Sides opposite to equal angles of a triangle are Angle opposite to equal sides of a triangle are In an equilateral triangle all angles are (iv) In a ABC if A = C, then AB = (v) (vi) If altitudes CE and BF of a triangle ABC are equal, then AB = In an isosceles triangle ABC with AB = AC, if BD and CE are its altitudes, then BD is CE (vii) In right triangles ABC and DEF, if hypotenuse AB = EF and side AC = DE, then ABC 22 Which of the following statements are true and which are false (i) Sides opposite to equal angles of a triangle are unequal (ii) Angles opposite to equal sides of a triangle are equal (iii) The measure of each angle of an equilateral triangle is /
11 (iv) (v) (vi) If the altitude from one vertex of a triangle bisects the opposite side, then the triangle is isosceles The bisectors of two equal angles of a triangle are equal If the bisector of the vertical angle of a triangle bisects the base, then the triangle may be isosceles (vii) If any two sides of a right triangle are respectively equal to two sides of other right triangle, then the two triangles are congruent (viii) Two right triangles are congruent if hypotenuse and a side of one triangle are respectively equal to the hypotenuse and a side of the other triangle 23 ABC and DBC are two isosceles triangles on the same base BC Show that ABD ACD 24 If ABC is an isosceles triangle with AB = AC Prove that the perpendicular from the vertices B and C to their opposite sides are equal 25 If the altitudes from two vertices of a triangle to the opposite sides are equal Prove that the triangle is isosceles 26 In a right angled triangle, one acute angle is double the other Prove that the hypotenuse is double the smallest side Inequalities in a Triangle (i) (ii) (iii) If two sides of a triangle are unequal, the longer side has greater angle opposite to it If two angles of a triangle are unequal, the greater angle has the longer side opposite to it The sum of any two sides of a triangle is greater than the third side /
12 IX ACADEMIC QUESTIONS Subjective Assignment4 1 Prove that the difference of any two sides of a triangle is less than the third 2 Show that of all the line segments that can be drawn to a given line from a given point not lying on it, the perpendicular line segment is the shortest 3 In a right angled triangle, prove that the hypotenuse is the longest side 4 In the figure, AD is the bisector of A, show that AB > BD 5 In figure PR > PQ and PS bisects QPR Prove that PSR PSQ 6 In the fig PQ > PR QS and RS are the bisector of Q and R respectively Prove that SQ > SR 7 In the fig x y, show that M N /
13 XI SCIENCE & DIP ENTRANCE Subjective Assignment (MCQ) 5 1 In the figure AB CD, 128 o 18 o 2 In the given figure AB CD, ABE 128, (b) 148 o 7 (b) (d) 13 o EFC 3 and 8 1 (d) 13 BED 2, what is the value of EDC? ECF 1, then BAF is equal to 3 In ABC, when BC is produced on both ways, the exterior angles are the value of A 36 (b) (d) and 134, what is 4 In the figure calculate the value of y if 45 (b) 5 6 (d) 9 5 In the figure the value of x is 1 (b) (d) 12 x 5 6 The angles of a triangle are 2x 1, x 2 and x 1 triangle it is? Equilateral Triangle Acute Angle Triangle (b) Right Angled Triangle (d) Obtuse Angle Triangle 7 In the trapezium ABCD, EF AD, what is the value of ACD? 7 (b) 5 (d) 6 4, which type of 8 In the figure CE is perpendicular to AB BDA? 5 (b) 7 (d) 6 9 ACE 2 and ABD 5, what is the measure of /
14 9 In the figure what is the relation of x in term of a, b and c? x a b c (b) x a b c 9 x 18 a b c (d) x 18 a b c 1 PN QR and PM is bisector of P in PQR, then ( Q R ) is equal to MPN (b) 2 MPN 1 MPN (d) 1 MPN The bisector of exterior angles B and C meet at O, what is BOC 12 (b) 8 45 (d) 4 12 The bisectors of exterior angles of ABC intersect at O and form a BOC which is always Acute Angle (b) Right Angle Obtuse Angle (d) None of these 13 The bisectors of interior angles of a triangle forms an angle which is always Acute Angle (b) Right Angle Obtuse Angle (d) All of these 14 In a triangle ABC, the internal bisectors of angles B and C meet at P and the external bisectors of the angles B and C meet at Q, then the BPC BQC is equal to 9 (b) 9 1/ 2 A 9 1/ 2 A (d) In the figure what is the value of a b c? 9 (b) (d) In the figure PM is bisector of P and PN is perpendicular on QR then the value of MPN is 3 (b) 6 (d) BM and CM are interior bisectors of B and C while BN and CN are exterior bisectors of B and C respectively Which is correct? (b) BMC 11 BNC 7 BMC BNC /
15 (d) All are correct 18 In a ABC, AB = 5cm, AC = 5cm and A = 5 o, then B = 35 o (b) 65 o 8 o (d) 4 o 19 If two sides of a triangle are unequal then opposite angle of larger side is greater (b) less equal (d) half 2 The sum of attitudes of a triangle is than the perimeter of the triangle greater (b) less half (d) less 21 In the given figure, PQ = QR, QPR = 48 o, SRP = 18 o, then PQR = 48 o (b) 84 o 3 o (d) 36 o 22 In the given figure, PQR is an equilateral triangle and QRST is a square Then PSR = 3 o (b) 15 o 9 o (d) 6 o 23 Can we draw a triangle ABC with AB = 3cm, BC = 35cm and CA = 65cm? Yes (b) No Can t be determined (d) None of these 24 Which of the following is not a criterion for congruence of triangles? SSA (B) SAS ASA (d) SSS 25 In the given figure, AB BE and EF BE Also BC = DE and AB = EF Then BD = FEC (b) ABD = EFC ABD = CMD (d) ABD = CEF 26 In quadrilateral ABCD, BM and DN are drawn perpendicular to AC such that BM = DN If BR = 8cm, then BD is /
16 4cm (b) 2cm 12cm (d) 16cm 27 In the given figure PQ > PR, QS and RS are the bisectors of Q and R respectively SQ = SR (b) SQ > SR SQ < SR (d) None of these 28 In the figure, PS is the median, bisecting angle P, then QPS is 11 o (b) 7 o 45 o (d) 55 o 29 In the given figure x and y are x = 7 o, y = 37 o (b) x = 37 o, y = 7 o x + y = 117 o (d) x y = 1 o 3 In the given figure BD AC, the measure of ABC is 6 o (b) 3 o 45 o (d) 9 o /
17 ANSWER Assignment 1 2 A = 71 o 5 3 o, 5 o, 1 o 6 52 o, 53 o, 75 o 9 18 o (b) 91 o 4 o, 4 o, 1 o (d) 18 o (e) one (f) one 1 False (b) True False (d) True (e) False (f) True Assignment 2 1, BED = 82 o 3 x = 6 o, y = 55 o 5 x = 115 o, y = 2 o 6 x = 4 o, y = 2 o 7 75 o Assignment (i) equal (ii) equal (iii) 6 (iv) BC (v) AC (vi) Equal to (vii) EFD 22 (i) False (ii) True (iii) True (iv) True (v) True (vi) True (vii) True (viii) True Assignment 5 1c 2d 3b 4b 5d 6b 7a 8b 9c 1d 11c 12a 13c 14d 15d 16a 17d 18b 19a 2b 21b 22b 23b 24a 25a 26d 27b 28c 29b 3d /
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