Statistics. To find the increasing cumulative frequency, we start with the first

Transcription

1 Statistics Relative frequency = frequency total Relative frequency in% = freq total x100 To find the increasing cumulative frequency, we start with the first frequency the same, then add the frequency of the next value to get the I.C.F. of the second value, then add the resulting I.C.F. to the frequency of the third value to get the new I.C.F. and so on. To construct the circle diagram, find the central angle of each value: Central angle = freq total x 360 Mean (average) denote by x: x = x1f1+xf. total frequecy

2 Radicals 0 =0 1 =1 a = a (a is a positive number) ( a) = a a x b = axb a b = a b a x a = a a and a are opposite numbers. a and b are inverses (reciprocals). b a To rationalize the denominator of a fraction we have to multiply by the conjugate of the denominator : a b x x x x = a x bx a c d c+ d x = a( c+ d) c + d c d

3 Trigonometry Sin α + cos α =1 In a right triangle (α is an acute angle). Sin α = opposite hypotenuse Cos α = adjacent hypotenuse Tan α= opposite adjacent sin α Tanα= cos α Let (d): y= ax+b If (d) is an increasing line, a > 0 then tan α = a where α is the acute angle made by (d) and x-axis. If (d) is a decreasing line, a < o then, tan α = -a To find the angle α if tan α is given α= tan 1 (a) shift tan (on calculator).

4 Circles tangents drawn from an external point to the same circle are equal, and the line joining the external point to the center is the : 1) angular bisector of the angles formed. ) perpendicular bisector of the segment joining the points of tangencies. (MA) and (MB) are tangents To circle (C), then: MA=MB (MO) is the angular bisector of AMB and AOB (MO) is the perpendicular bisector of [AB] The tangent is perpendicular to the radius at the point of tangency. AMB =90 (inscribed angle facing diameter)

5 In a right triangle, the median relative to the hypotenuse equal half of it. Meeting point of heights: orthocenter Meeting point of medians :centroid (center of gravity) Meeting point of angular bisectors: in center Meeting point of perpendicular bisectors: circumcenter.

6 Vectors Two vectors are equal if they have same sense, direction and magnitude. X =X B -X A AB y = y B - y A AB If vectors are equal, then they have same coordinates. If B is the translate of A by vector u AB =u AB +BC =AC (Chasle s Relation) AB + AC =AD where D is the fourth vertex of the parallelogram formed by [AB], [AC].

7 Inequalities A first degree inequality is of the form ax+b 0, ax+b 0 ax+b>0, ax+b<0 ax+b 0 x b a ax+b 0 x b a ax+b > 0 x > b a ax+b < 0 x< b a Note: In equalities, when dividing or multiplying by a negative number, the inequality sign is reversed.

8 Power Rules a 1 = a a 0 = 1(a 0) a n = a a a a.. a (n times) a m a n = a m+n = am n an (a m ) n = a m.n (a b) n = a n b n ( a b )n = an b n a n = 1 a n (a + b) n a n +b n False (a b) n a n - b n False a m Scientific notation: A number is in scientific notation if it is of the form a 10 n where a is an integer 1 a 9 and n is an integer.

9 Pythagoras theorem If triangle ABC is right at B: AC =AB +BC Converse of Pythagoras theorem: If in a triangle, AB +BC =AC then triangle ABC is right at B. Semi equilateral triangle: (90,60,30) Leg facing 30 = hyp Leg facing 60 = hyp 3 In an isosceles triangle, the median issued from the vertex is the same as the height, angular bisector, perpendicular bisector. Rational number: is any number of the form a b integers. (b 0) where a and b are Irrational: any number that can t be written in the form a (the decimal b part is unlimited) Ex: π, 5, 7.. Decimal fraction: a where b is a power of 10 b Ex: 7 10,1, 3 5 (if the denominator is multiple of, 5, or both it will be decimal fraction) A decimal number is a number where the decimal part is limited.

10 Remarkable Identities (a + b) = a + ab + b (a b) = a ab + b (a+b) (a-b)= a b Notes: (-a-b) =(a+b) (a-b) =(b-a) To solve an equation = 0, take the factorized form, then solve each one of the factors = 0. To solve an equation = constant, take the expanded form = constant. To find the domain of definition of a fractional expression, put the denominator 0 then solve: ax+b cx+d, we have cx + d 0 x d c

11 Midpoint theorem: If M is the midpoint of [AB] and N is the midpoint of [AC] then (MN) and (MN) = BC (BC) midpoint theorem If M is the midpoint of [AB] and (MN) (BC), then N is the midpoint of [AC] (converse of midpoint theorem). To find the height in a right triangle, write its area in different ways, then put them equal to each other that is: Area of ABC = AB AC Area of ABC = AH BC Then AB x AC= AH xbc AB, AC, BC are given Find AH

12 Similar Triangles How to prove similar Triangles: 1) equal angles ) 1 equal angle, and the respective sides including the angles are proportional Ex: If A = D and AB DE = AC DF 3) the 3 sides are respectively proportional when triangles are similar, then their respective sides are proportional which is called (ratio similarity). To find this ratio similarity, put the triangles with their equal angles under each other: DEF ABC : AB DE = BC EF = AC DF

13 Areas of some shapes: Area of a square = S Area of a rectangle = ι w Area of a parallelogram =h b Area of a rhombus = d1 d Area of a trapezoid = h(b1+b) Area of a triangle = h b AMB = AB (inscribed angle) AOB =AB (central angle)

14 To prove a table of prpoprtionality Proportionality a c b d We prove a c = b d A raise or increase by d% new value = old value (1+d%) A discount or decrease by d% New value = old value (1-d%) When X and Y are proportional, then one of them is a linear function of the other, then the algebraic expression is y = ax The graph of a linear function is a straight line passing through the origin.

15 Thales s theorem (MN) (BC), then: AM AB =AN = MN AC BC AM = AN MB NC Converse of Thale s Theorem: IF AM AB =AN, then (BC) (MN) AC If AM AB = AN AC = MN BC = k a) o< k<1 AMN is the reduction of ABC of center A and ratio=k. b) k>1 AMN is the enlargement of ABC of center A and ratio= k. Lines in a coordinate system: (d): y = ax + b a: slope (direct coefficient) b: y- intercept To find x- intercept, put y = 0 then find x To find y-intercept, put x =0 then find y If a line parallel to x-axis :y = k If a line parallel to y-axis x = k

16 If lines (d) and (d ) are parallel, slope (d) =slope(d ) If lines (d) and (d )are perpendicular slope (d) x slope (d )= -1 (their slope are opposite,reciprocal) Slope (AB) = yb ya xb XA. AB= (xb xa) + (yb ya) If a line passes through origin (d) :y=ax (b=0) If a point A belongs to a line (d) then the coordinate of A satisfy the equation of the line (d): ya = ax A +b I is the midpoint of [AB]: x I = XA+XB YA+YB y I = In a right triangle, the center of the circle circumscribed is the midpoint of the hypotenuse, and the hypotenuse is the diameter, R= hyp To prove (d) is the perpendicular bisector of [AB]: 1 - (d) (AB). - (d) passes through midpoint of [AB]. Any point lying on the perpendicular bisector is equidistant from extremities Any point lying on the angular bisector is equidistant from the side of the angle If A is the symmetric of B with respect to point I,then I is the midpoint of [AB] To find the point of intersection of lines (d) and (d ),put their equations as a system of equations in unknowns, then solve to find x and y (easiest way by comparison, that is put y=y, the solve to find x)