4 5 6. Bs a 9. 10. uw : (side? 31/2 (D) (Ps#P2) :LxB : Base x Altitude 21/2 (De#D2) 2 xar iat 21/2 Base x Altitude 21/2 Altitude x (sum of // sides) Area of a Trapezium +: V3/4 a? (where aside of the triangle) Area of an equilateral Triangle ‘Area of a Triangle by hero's formula 2 Vs(sa) (S-b)(5-¢) where, szatbtc o Volume’s of Solid Figure’s Volume of a Cube = side? Cu. Units Volume of a Cuboid = Ix b x h Cu. units Volume of a Cylinde h Cu. Units Volume of a Cone = 4 ar“h cu. units 3 Volume of a Sphere = 4 xr cu. units Volume of a hemisphere = 2 nr’ cu. units 3 Volume of a frustum of a cone = Anh(r, + 7, +f, f2) 3 Curved Surface area’s of Solid Figures C.S area of a cube 4s? sq. unit CS area of a cuboi (1+ b) h sq. unit C.S area of cylinder = 2xr (r + h) sq. unit CS area of a Cone = xrl sq. unit where I= Vh? +? CS area of Hemisphere = 2 r? sq. unit CS area of frustum of a cone = a(r; + r2) where |= Vh? + (ry-r2)? Surface area’s of Solid Figures geecan not have more than one common between them Ifa straight line stands on another straight line , then the sum of two adjecent angles is equal to 180 If the sum of the two adjacent angles is equal to 180 degrees, then their exterior arms are int he same straight line. If there is a given line manda given point p outside it then one and only one line can be drawn parallel to the given line through the given point outside it Iftwo straight lines intersect, then the vertically opposite angles thus formed are (if CD stands on AB) (then 21+ 22 180") Cc A * B af Z1+22= 180°) (then AB & BD are in the) (same straight line Pe 8 (£1=22,23=24)cutting lines is to 180 degrees the lines are parallel. outside it then one and only one perpendicular can be drawn to the given line L through the I given point p. t im i p ‘3 LH. _Ifthere are three or more a than three parallel lines and b if they make equal intercepts . on one transversal, then they / will make equal intercepts on another transversal as well (if PQ = QR) (then ‘ST=TM) 1 RIANGLES TRIANGLES 12. The sum of the three angles of a triangle is equal to 180 degrees. 2 - (2 1+ 2 2+2 3=180+) (Angle sum property of a triangle 13. The exterior angle of a triangle is equal to the sum of the two interior opposite (4 I= 2 2+2 3) angles. Exterior Anglg property 14. _ Ina triangle equal sides have equal angles opposite to them.Ina triangle greater angle has longer side opposite it. Ina triangle the sum of two sides is always greater than the third. In a triangle the line joining the mid points of two sides of a triangle is parallel to the third and equal to half of it. Ina triangle the line drawn from the mid point of one side of the triangle parallel to the second side bisects the third side ar AB is greater than AC) (then Z C is greater than 2B B (if Z Cis greater than 2 B) (then AB is greater than AC) B AB + BC is greater than CA A B (if AD= DB and AE = EC) (then DE parallel BC) (and DE = 1/2 BC) A (If AD= DB and if DE is) parallel to BC then AE=EC)ifand only if: S.A.S = S.A.S. AS.A=A.S.A. Two triangles are said to be similar:- If they are equiangular If their sides are proportional i ZA=2D, 2 B=2Z Eand 2 C=2 F, then triangle ABC and DEF are similar If AB/DE = BC/EF = CA/FD then ABC is similar to DEF If one angle of one a ‘ triangle is equal to one angle of the other © . triangle and the sides containing the equal it Z A= Z D and AB/DE = AC/DF then the angles are proportional. triangles ABC & DEF are similar The Medians of a triangle meet at centre and the centroid divides the median in the ratio 2:1 The right bisectors of the sides of a triangle meet at the ciroum-cente. ‘The altitude of a triangle meet at ortho centre. a) Centroid is the point where medians of a trangle meet. b) The incentre is the point where the angle bisectors of ; erateA triangle is scalene if all its sides or all its angles are different from each other. A triangle is right angled if one of its angles is 90" The side opposite to 90 degrees is called Hypotenuse A triangle is said to be acute angled if all its angles are less than 90 degrees. A triangle is said to be obtuse angled if one of its angles is greater gled triangle the square on the WAB=AC orlf 2B= ZC iangle ABC is an iso 8 WAB#BC #4 CAorif ZA42B42C then triangle ABC is sealene 5. a If ZC = 90 degrees then triangle ABC is rt angled triangle If ZA = less than 90 d 2B = less than 90 degi ZC = less than 90 degre the triangle ABC is acute angled triangle. o + than 90 degrees then triangle ABC is obtuse angled triangle A 8 Gopposite angle are equal. The diagonals bisect each other. A parallelogram is a rectangle if and only if its diagonals are of equal length. A parallelogram is a thombus if and only if its diagonals are perpendicular A parallelogram is a square if and only are equal and perpendicular. IRCLES Acircle is the locus ofa point which moves in such a way that it is equidistant from a fixed point called the centre of the circle. Coe Xow He Quad, ABCD is a parallelogram 18 ZA= £0 & £B= 2D J. ABCD is a parallelogram )=O0C & BO= OD gram ABCD is a Rhombus if AC perpendicular BD. Lae Parallelogram is a square if and only if AC = BD and AC perpendicular BDArea of a circle is xr’ Circumference of a circle is 2nr 46. Two circles are congruent if and only if they are of equal radii. 47. Inacircle equal chords subtended equal angles at \ the centre and conversely \ if two chords subtended equal angles at the centre, IfAB=CD then 2 AOB = ZCOD then the chords are equal ‘& Conversely if ZAOB = ZCOD then AB =€D 48. Ifa perpendicular is drawn from i saa ee hte ° a chordnd conversely the Ye ‘oie line joining the centre with THe the mid point of the chord is if ZP = 90 degree then AP = PB & conversely iPAP. perpendicular to the chord PR then 2P ~ 90 degree 49. Two ares ofa circle are congruent if they subtended equal angles at the centre and conversely. esis ifare ABC Two arcs ofa circle are congruent chords are56. 37. 58. If there are four non-collinear points and if the line joining any two points subtends equal angles at any other two points on the same side of it then the four points are concylic Two and only two tangents can be drawn to a circle from a point outside it and the two tangents are equal. Ina circle, the radius through the point of contact is perpendicular to the tangent. Ina circle the angle that the chord through the point of contact subtends with the tangent is equal to the angle subtended by the chord in the alternative segment and conversely If two circles touch each other, the line joining the centres is perpendicular to the common 7a = ZC then AB CD are concyclic om ae AP=PB Gs 2P-= 90 degrees @62. The area ofa parallelogram is the product of any of its sides and the corresponding altitudes. Triangles on the same base and between the same parallels are equal in area. The area of a triangle is half the product of any of its sides on the corresponding altitudes. Ifa triangle and a parallelogram are on the same base and between the same parallels, the area of the triangle is equal to half that of the parallelogram. ‘The area of a trapezium is half the product of its height and the sum of the parallel sides. ar (Triangle PBC) = ar. (Triangle ABC) pe ar, (Triangle PBC) =1/ BC x AL apes “ss fe ar, (Triangle EAB) = 1/2 ar. ( Parallelogram ABCDALGEBRAIC IDENTIT Be (xb) = x Ha+b)x+ab 2. a+b)" ) ei Cee 3 @oy ea cab 4 abt (a+b)(a-b) 5. (a+b+c) a+b’ +c? + 2ab + 2ca 8. (a+ by a’ + b’ + 3ab (a + b) % @-b) a’ - b’ -3ab (a - b) ae Mare ph: (a-b) (a’ + ab + b’) 9 ap = (a+b) (a’- ab +b’) 10. a+b +0°- 3abc = (a+b + c)(a’ +b’ +c’ -ab - be - ca) Ifat+b+c=0 then a’ +b? + c’- 3abc = (0)(a? + b +c? - ab - be - ca) => a'+b’+c°-3abc=0 => a'+b'=c° = 3abe 1. Distance Formula Q(x,, Ye) Distancebwpq = PApy) Os VG=x) + Ye- VF A(x1,y,). om, om, Box, y.) 2. Section Formula ue yy ; P(x, y) d The Co-ordinates of the point P(x, y) which divides the line segment joining the pts A (x1, y1) & b(x2, y2) internally in the ratio m1 : m2 are ( mx. +My, my, + my, ) m+m, | om,+m, The mid points of the line segment. Joining the points A (x1, y1) & B (x2 y2) If point ‘P' divides AB in the ratio 1: 1 then the co-ordinates of the point ‘P’ will be xc CS Area of the Triangle : Formed by the pointA (x,, y,), B(x, y,) Atm & C(x3,y3) in the numerical values of the expression is V/s (Yar Ys)#%2(Ys-¥s)+Xs(Y1-Y2) $q. unitsCosec (90° - 8) = Sec @ Tan (90° - 8) = Cot 6 Cot (90°-8) = Tan 6 Trigonometric Identities Sin’ @ + Cos’ 8 = 1 Sin’ 8 = 1 - Cos’ Cos’ 8 = 1 - Sin’ @ 1+ tan’ 6 = Sec’ @ tan’ @ = sec’ 0-1 or Sec’- 1 = tan’ Sec’@ - tan’ = 1 Cot’e + 1 = Cosec’ 8 Cot’ 6 = Cosec? 8-1 or Cosec’ @- 1 cot’ 8 Cosec’ @ - Cot’ @=1