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Solved Geometry problems for competitive exams
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  Prepared By  Harshwardhan Phatak  Page 1 of 54Quant Funda Class (Geometry) a dcb ANGLES and PARALLELS (a)Two straight lines which meet at a point form an angle between them. Acute angle: 0° < a < 90° Right angle : b = 90° Obtuse angle : 90° < c < 180°Reflex angle : 180° < d < 360° (b)Theorems :If AOB is a straight line, thena + b = 2 right angles (Adjacent angles on a straight line) If a + b = 2 right angles, thenAOB is a straight line. (Adjacent angles are supplementary) The sum of all the angle at a point, eachbeing adjacent to the next, is 4 right angles.   a + b + c + d + e = 4 right angles (Angles at a point) If two straight lines intersect, thevertically opposite angles are equal.a = b, c = d (Vertically Opposite angles) babaadbcdeabc Plane Geometry  Prepared By  Harshwardhan Phatak  Page 2 of 54Quant Funda Class (Geometry) Parallel lines PQ and RS are cut by a transversal LM, then we have :1.The corresponding angles are equal c = b (Corresponding angles, PQ || RS) 2.The alternate angles are equal a = b (Alternate angles, PQ || RS) 3.The interior angles are supplementaryb + d = 2 right angles (Interior angles, PQ || RS) If PQ and RS are cut by transversal LM, thetwo lines are parallel if a = b (Alternate angles) or c = b (Corresponding angles) or b + d = 2 right angles (Interior angles are supplementary) (c) (i)Two angles whose sum is 90°, are complementary.Each one is the complement of the other.(ii)Two angles whose sum is 180º, are supplementary. Each one is the supplement of the other. TRIANGLES PROPERTIES : 1.Sum of the three interior angles is 180° 2.When one side is extended in any direction, an angle is formed with another side. This iscalled the exterior angle. 3.There are six exterior angles of a triangle. 4.Interior angle + corresponding exterior angle = 180°. 5.An exterior angle = Sum of the other two interior angles not adjacent to it 6.Sum of any two sides is greater than the third side. 7.Difference of any two sides is less than the third side. 8.Side opposite to the greatest angle will be the greatest and vice versa. 9.A triangle must have at least two acute angles.10.Triangles on equal bases and between the same parallels have equal areas.11.If a, b, c denote the sides of a triangle then (i)if c² < a² + b², Triangle is acute angled (ii)if c² = a² + b², Triangle is right angled(iii)if c² > a² + b², Triangle is obtuse angled  IMPORTANT POINTS WITH RESPECT TO A TRIANGLE : a. Centroid : When a vertex of a triangle is joined to the midpointof the opposite side, we get a median. The point of intersection of the medians is called the CENTROIDof the triangle. The centroid divides any median inthe ratio 2 : 1. PQRSLM cadb ABCDEGABCD  Prepared By  Harshwardhan Phatak  Page 3 of 54Quant Funda Class (Geometry) Any median of a triangle bisects the area of the triangle. In the figure AD is the median from A.and BD = CD We have the formula : 2 x (Median)² + 2 x (½ Third side)² = Sum of squares of other two sides  2 x (AD)² + 2 x (½ BC)² = AB² + AC².Area of triangle ABD = Area of triangle ADC. If BD = DE = EC then the areas of trianglesABD, ADE, AEC are equal. D & E are called the points of trisection of BC. b. Circumcentre : The point at which the perpendicular bisectors of the sides of a triangle meet is the circumcentre of the triangle. The circumcentre S of a triangle isequidistant from the three vertices. We have SA= SB = SC = circumradius.The circle with center S and passing through A,B& C is called the circumcircle of triangle ABC.In the triangle : BSC = 2    BAC,   ASB = 2    ACB,    CSA = 2    ABCAny point on the perpendicular bisector of a line isalways equidistant from the ends of the line. c. Incentre : The point of intersection of the angle bisectors of a triangle is called the incentre I. The perpendiculardistance of I to any one side is inradius and thecircle with centre I and radius equal to inradius iscalled the incircle of the triangle. The three sidesare tangent to the incircle. Any point on the bisectorof an angle is equidistant from the arms of theangle. The incentre divides the bisector of     A inthe ratio (b + c) : a.Also    BIC = 90 + A/2,    AIB = 90 + C/2,   AIC = 90 + B/2.The bisectors of two exterior angles at B & C andthe bisector of     A meet at a point called excenter I’. There can be three ex–circles of a triangle.Also I’E = Ex–radius. There can be three ex–radii to a triangle.    BI’C = 90 – A/2. NOTE : (a) Angle bisector Theorem :In the figure if AD is the angle bisector (Interior)of     BAC then1.AB/AC = BD/DC2. AB x AC – BD x DC = AD²  AB CS . I'BICcbaDrA  AB CD  Prepared By  Harshwardhan Phatak  Page 4 of 54Quant Funda Class (Geometry) (b)In the figureIf AD is the bisector of exterior angle at A of triangleABC, then : AB/AC = BD/DC(c)In the figure, if in triangle ABC, AN is thebisector of     BAC and AM is perpendicular toBC then    MAN = ½ (   B –     C).(d)In a triangle ABC, if BC is produced to D and AL is the bisector of     A,then    ABC +    ACD = 2   ALC(e)In triangle ABC, if side BC is produced to D and bisectors of     ABC and   ACD meet at E, then    BEC = A/2 d. Orthocentre : The perpendiculars drawn from vertices to opposites (called altitudes) meet at a point calledorthocentre of the triangle.Also    BOC = 180 –     A   AOB = 180 –     C   AOC = 180 –     BArea of triangle = ½ x Base x Altitude.  NOTE :  There is similarity between the topic on geometry & that on mensuration. You arerequested to go through this topic only after thoroughly grasping the chapter on mensuration as formulae and facts discussed in that chapter are not repeated here. Some Important Points :1.Isosceles triangle : In this the base angles (or any two angles are equal).The bisector of     A is perpendicular bisector of thebase and is also the median to the base. 2.Equilateral Triangle : (a)All the four points viz. centroid, circumcentre, incentre, orthocentre coincide.(b)Medians, angle bisectors, altitudes, perpendicular bisectors of sides are all represented bysame straight lines.(c)Given the perimeter, equilateral triangle has the maximum area.(d)Of all the triangles that can be inscribed in a circle, the equilateral triangle has the greatestarea. Points (c) and (d) give you a hint regarding the nature of symmetry. 3.Right triangle : Median to the hypotenuse = ½ x hypotenuse = circumradius AEBDC  AB CM N AB CODCB A AB=AC
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