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FORMULA AT A GLANCE Amount x R x T 1. True discount 100 + (R x T) 2. Tan 150 = 2- VS 3. When Two points are on the same side Of a pole with angle of elevations given, then the distance between the points .600 .B 4. Cyclic parallelogram is either a square or a rectangle 2 = E-L cot 5. Area of a regular polygon of 'n' sides 4 n 1 Volume of a pyramid - x base x height 6. 3 7. Base area of a regular hexagon = 2 8. X radian = 1800 9. 1+2+3+...+ n = 2 12-1- 22+ 32 = 2 where a = side 11. The sum S of the first n numbers of an arithmetic progression is given by the formula: S = E(al + an) where al is the first term and an the last one. S = - [2m + d(n-l)] 2 12. In General we write a Geometric Sequence like this:
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3 {a, ar, ar , ar , 13. Summing a Geometric Series When we need to sum a Geometric Sequence, there is a handy formula. (n-l) To sum: a + ar + ar + + ar Each term is ar , where k starts at 0 and goes up to n-l n—l Use this formula: I—r a is the first term r is the "common ratio" between terms n is the number of terms If a side of a cyclic quadrilateral is produced then the exterior angle is equal (12) to the interior opposite angle. 2 c 1 Ll=L2 (3) If two chords AB and CD intersect internally or externally at point P then. PA x PB = PC x PD (4) If PAB is a secant which intersects the circle at A and B and PT be a tangent at T, then B = PA x PB (5) The angles which the chord makes with the tangent line are equal respectively to the angles formed in the corresponding alternate segment.
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c 2 1) (1) If two tangent PA and PB are drawn from the external point P, then Ll=L2 and L3=L4 OP is perpendicular to AB and AC=BC rl and r2 are the radius of two circles and d is the distance between the centres (2) of the circle then the length of the direct common tangent of two circles is given by (3) If rl and are the two radius of the circle and "d" is the distance between them then the length of the transverse common tangent is given by x If a circle touches all the four sides of a quadrilateral then the sum of opposite pair of (4) sides are equal. i.e. AB+CD=AD+BC (5) If two chords AB and AC of a circle are equal then the bisector of LBAC passes through the centre O of the circle. 2 c (6) The quadrilateral formed by the angle bisector of a cyclic quadrilateral is also cyclic.
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(9) If two circles of same radius r are such that the centre of one lies on the circumference 1= 43 x r of the other then the length of the common chord is given by o (10) If 2a and 2b are length of two chords which intersects at right angle and if the distance between the centre of the circle and intersecting point of the chords is C then the radius of circle is given by NY O 2 (11) If three circles of radius r are bound by a rubber band then the length of rubber band is given by = 6r+27tr In-centre: The point of intersection of the all the three angle bisectors of a triangle is called as In-centre c D The distance of the in-centre from the all the three sides is equal (ID=IE=IF=inradius "r") In-radius (r)= Area of triangle/Semiperimetre=MS LB1c = 90 +LA/2 Circumcentre: LAIC=90+LB/2 LAIB=90+LC/2 The point of intersection of the perpendicular bisectors of the three sides of a triangle is called its circumcentre.
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b c The distance between the circumcentre and the three vertices of a triangle is always equal. OA=OB =OC=R (circumradius) = abc/4A LBOC=2LA Orthocentre: LAOC=2zB LAOB=2zC It is the point of intersection of all the three altitudes of the triangle. c LAHB = 180 - LBHC = 180 - LA Centriod: LAHC = 180 - LB It is the point of the intersection of the three median of the triangle. It is denoted by G. A centroid divides the area of the triangle in exactly three parts. 2 1 2 1 c Medians: Median bisects the opposite side as well as divide the area of the triangle in two equal parts. Some important tricks are as follows: (1) In a right angle triangle ABC, LB=900 & AC is the hypotenuse of the triangle. The perpendicular BD is dropped on the hypotenuse AC from the right angle vertex B,
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c 1/BD2 = 1/AB2 +1/BC2 In a triangle ABC, AE, CD and BF are the medians then 3(AB2+BC2+AC2) = 4(CD2+BF2+AE2) Triangles and their properties 1. When all sides are given (Heron's formulae): Area = S = (aq-b+C)/2 3. When two sides and corresponding angle is given.: 1/2 a x c x sino b) (s - c)} where, 4. When all the median are given (Median is a line joining the vertex to the opposite side at midpoint): 4/3 x {s(s - ml) (s - m2) (s -m3)} 5. When all the heights are given: 1/Area of A = 4 {G (G — l/hl) (G — 1/112) (G — 1/113), Note: Where, G = 1/2 (1/111 + 1/112 + 1/113) Sine formulae of triangle a/SinA = b/sinB = c/SinC = 2R Where, R is Circumradius Cosine Formulae of triangle CosA — CosB= cosc - 2 — a2 / 2bc — b2 / 2ac 2 — c2 / 2ba
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An interesting result based of cosine formulae If in a triangle CosA = b2 +c2 — a2 / 2bc, whereas b & c are the smaller sides then Case I, b2 +c is greater than a then angle A is acute. Case Il, b2 +c is smaller than a then angle A is obtuse. Case Ill, b2 +c2 is equals to a2 then angle A is Right Angle. For example TRIANGLES AND ITS CERVICES Medians of a Triangle c . The medians always intersect in a single point, called the centroid. Properties: 1. Centroid divides the Median in the ratio 2:1. i.e Al/ID = 2/1 2. Apollonius Theorem:- To find the length of median when all sides are given: 4 x AD2= 2(AC2 + AB2) - BC2 3. All 3 median of a triangle divides the triangle in 6 equal parts. Angle Bisectors All the three angle bisectors meet at a common point called as Incentre. Properties: 1. Inradius = Area of A AIA2A3 / Semi perimeter of A AIA2A3 2. Angle formed at the incentre by any two angle bisector: Angle AllA3 = Angle AIA2A3 /2) + 900
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Perpendicular Bisectors of Sides of the Triangle When the perpendicular bisectors of the side of the triangle is drawn they meet at a common point known as circumcentre. Properties: In the adjoining figure DP, EP, & FP are perpendicular bisector of sides of the triangle. Point P is circumcentre. 1. Circumradius = length of side AC x CB x BA / 4 x Area of triangle 2. Angle formed at the circumcentre by any two linesegment joining circumcentre to the vertex: Angle APC = 2 x Angle ABC Altitude of the Triangle c The orthocenter of a triangle is the point where the three altitudes meet. 1. Angle BOC + Angle BAC = 1800 SIMILAR TRIANGLE If two triangles are similar, then the corresponding sides we have, shows the following relation: The ratio of sides of triangle is proportional to each other. The height, angle bisector, inradius & circum radius are proportional to the sides of triangle.
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Median AABC / Median ADEF = Height AABC/ Height ADEF = ABIDE 2 Area AABC/ Area ADEF = (ABIDE) IAnQA ABC/ if the bisectors ofZBan&ZCmee€at O them c ZBOC= ZA 2 2. In a AABC, if sides AB and AC are produced to D and E respectively and the bisectors of ZDBC and zfCB Intersect at O, then x O c Z BOC = ZBAE = D c 1/2 CZABC- ZACB) In a AABC, If BC is Produced to D and AE IS the Angle isector of LA, then c ZABC and ZACD = 2 ZAEC
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In a right angle AAB angle vertex B, then 3 B = 90 and ACIS hypotenuse. The perpendicular BD is dropped on ypotenuse AC from righ c (l) BD = (AB x BC)/ (AC) (ii) AD = AB2 /AC (iii) CD = BC2/AC (iv) 1/BD2 = (1/AB2) + (1/BC2) Parallelogram: Area of parallelogram is double the area of the triangle formed by diagonals. Bisectors of the angle of a parallelogram form a rectangle. 4 4 p, s p 3 1 2 2 A parallelogram inscribed inside a circle is rectangle. A parallelogram circumscribed about a circle is a rhombus. Area of parallelogram ABCD= Base x Height A parallelogram is a rectangle if its diagonal are equal. Rhombus: D c 900 a AC2+BD2=4AB2 Convex Polygon: A polygon in which none of its interior angle is more than 1800. Concave Polygon:A polygon in which at-least one of its interior angle is more than 1800
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Regular Polygon: Number of diagonals =n(n-3)/2 Normal equation of line = ax + by + b) Slope — Intercept Form Where, m = slope of the line & c = intercept on y-axis Solution: Area of triangle is = 1/2 * x-intercept * y-intercept. Trigonometric form of equation of line, ax + by + c = 0 x cos 0 + y sin 0 = Where, cos - a/ ((a2 + b2) , sin 0 - = b/ Al(a2 + b2) & p = C/N(a2 + b2) P, Equation of line passing through point (XI,YI) & has a slope m = y— = m (x - Xl) 4. Slope of line = — Yl/X2 — Xl = Angle between two lines Tan e = ± (nu — num2) where, ml , nu = slope of the 5. lines 6. Equation of two lines parallel to each other ax + by + Cl = 0 ax + by + c2 = 0 7. Equation of two lines perpendicular to each other ax + by + Cl = 0 bx — ay + c2 = 0 8. Distance between two points (Xl, YD, (x2, Y2) D = (X2 — Xl)2 + (Y2 — Yl)2 9. The midpoint of the line formed by (Xl, YD, (x2, Y2) M = (Xl + X2)/2, (Yl + Y2)/2 Area of triangle whose coordinates are (Xl, YD, (x2, h), (x3, Y3) Some useful Short trick: If there is a change of X % in defining dimensions of the 2-d figure then its perimeter will also changes by X %
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If all the sides of a quadrilateral is changed by X % then its diagonal will also changes by X%. 2. The area of the largest triangle that can be inscribed in a semi circle of radius r is r If the length and breadth of rectangle are increased by x % and y % then the area of the rectangle will increased by. (x + y + xy/100)% If the length and breadth of a rectangle is decreased by by x % and y % respectively then the area of the rectangle will decrease by: (x + y - xy/100)% Numerically angle of elevation is equal to the angle of depression. Both the angles are measured with the horizontal. Angle Sin Cos 1 Tan d= h (cot 01 — cot 02) 300 313/2 450 1/V2 1/1/2 1 600 V3/2 900 1 CO 1 B 450 1 D 40 m Frustum of a Cone r = top radius, R = base radius, h = height, s = slant height Volume: V = T/ 3 (r2 + r R + R2)h Surface Area: S = s(R + r) + 7tr + TR Pyramid 2
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Volume of a right pyramid = (1/3) x area of the base x height. Area of the lateral faces of a right pyramid = (1/2) x perimeter of the base x slant height. If the difference between CI and SI for two years at a particular rate is given then 100 2 x Difference Principal — If the difference between CI and SI for three years at a particular rate is given then 1003 x Differenc Principal — r+300 If the ratio of the speeds of A and B is a : b, then the ratio of the Or b : a. the times taken by then to cover the same distance is Suppose a man covers a certain distance atx km/hr and an equal distance aty km/hr. km/h I 2xy Then, the average speed during the whole journey x+ If the speeddownstream is a km/hr and the speed upstream is b km/hr, then: Speed of boat in still water = 1 (a + b) km/hr. 2 Rate of stream = — (a — b)km/hr Rule 1: A man can row certain distance downstream in tl hours and returns the same distance upstream in t2 hours. If the speed of stream is y km/h, then the • still water is given by t) km/h in Rule 2: A man can row in still water at x km/h. In a stream flowin at km/h, if it takes him t hours to row to a place and come back, then the distance between two places is given by t(x2 - y2) 2x
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stream flowing aty km/h, if it takes t Rule 3: A man can row in still water at x km/h. In a hours more in upstream than to go downstream for the same distance, then the distance is given by t(x2 - y2) 2y Rule 4: A man can row in still water at x km/h. In a stream the same distance up and down the stream, then Upstream x Downstream Man's Speed in still water owin aty km/h, if he rows is given by x km/h x An important relation for allegation is:- Quantity of cheaper _ C.P. of dearer Mean Price Quantity of dearer — Mean price C.P. of cheaper
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