STATISTICS

SURFACE AREAS AND VOLUMES

PROBABILITY

PROBABILITY

                                                                                     

IMPORTANT POINTS TO REMEMBER

·      A event for an experiment is the collection of some outcomes of the experiment.

·       Empirical probability P(E) of an event E is given by

·      P(E) = Number of trials in which E has happened

         Total number of trials

·      Probability of an event lies between 0 and 1 ( both inclusive).

·       Sum of the probabilities of all the events of an experiment is 1. 

HERON'S FORMULA

HERON'S FORMULA

IMPORTANT POINTS TO REMEMBER

·       Area of a triangle =  1 /2* base  * corresponding height.

·       Area of a right angled triangle=  1 /2 ab, where a and b are the sides adjacent to right angle.

·        Area of an equilateral triangle of side a =√3/4 a².

·       Area of a trapezium =  1 /2   * (sum of its parallel sides)* distance between parallel sides.

·        Heron's Formula:

Area of a triangle =     s(s-a)(s-b)(s-c)

where s= semi perimeter and a, b, c are the sides of the triangle.

·       Perimeter of any triangle=sum of the length of its sides.

·       Perimeter of a rhombus of side a =4a.

·       Area of a polygon can be calculated by dividing the polygon into triangles and using Heron's formula for calculating area of each triangle.

CONSTRUCTIONS

CONSTRUCTIONS

IMPORTANT POINTS TO REMEMBER

·       To draw geometrical figure accurately some basic geometrical instruments are needed.

                                                      i.     A graduated scale, on whose one edge marked centimeters  and millimeters and on the other side inches and their parts.

                                                     ii.    A pair of set-squares, one with angle 60° and 30° and other with angles 45° and 45° .

                                                    iii.    A pair of dividers (or divider) with adjustments.

                                                   iv.     A pair of compasses (or a compass) with provision of filling a pencil at one end.

                                                     v.     A protractor.

Normally, all these instruments are needed in drawing a geometrical figure, such as a triangle, a circle, a quadrilateral, a polygon etc. with given measurements. But a geometrical construction is the process of drawing a geometrical  figure using only tow instruments an use graduated ruler, also called a straight edge and compass. In construction where measurements are also required you may use graduated scale and protractor also.

CIRCLES

CIRCLES

IMPORTANT POINTS TO REMEMBER

·       The collection of all the points in a plane, which are at a fixed distance form a fixed point in the plane, is called a circle.

The fixed point is called the centre of the circle and the fixed distance is  called the radius of the circle .

·       A circle divides the plane on which it lies into three parts. They are(i) inside the circle, which is called the interior  of the circle; (ii) the circle and  (iii) outside the circle, which is also called the exterior of the circle. The circle and its interior make up the circular region.

·       A line segment joining any two points on a circle is called a chord of the circle.

A chord passing through the  centre of a circle is known as its diameter. A diameter is the longest chord and its length is equal to twice the radius.

·       A piece of a circle cut between two points is called an arc. There are two, one longer and other smaller .The larger one is called the major arc PQ and the smaller one is called minor arc PQ. The minor arc PQ us also denoted by PQ arc and the major arc PQ bt PRQ, where R is some point on the arc between P and Q. Unless otherwise stated arc PQ and PQ stand for minor arc PQ. When P, Q are ends of a diameter, then both arcs are equal and each  is called a semicircle.

·       The length of the complete circle is called its circumference. The region between a chord and its corresponding arc is called a segment of the circle. There are two types of segments, major segment and minor segment. The region between an arc and the two radii, joining centre to the end points of the arc is called a sector. Like segments you find that minor arc corresponds to minor sector and major arc corresponds to major sector. The region OPQ is the minor sector and remaining part of the circular region is the major sector. When two arcs are equal that is each is a semi-circle, then both segments and both sectors become the same and each is known as semi- circular region.

·       If the angles subtended by two chords of a circle(or of congruent circles ) at the centre( corresponding centres) are equal, the chords are equal.

·       The perpendicular from the centre of the circle to a chord bisects the chord.

·       The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.

·       There is one  and only one circle passing though three non-collinear points.

·       Equal chords of a circle (or of congruent circles) are equidistant from the centre(or corresponding centres).

·       Chords equidistant from the centre (or corresponding centres) of a circle( or of congruent circles are equal.

·       If the two arcs of a circle are congruent, then their corresponding chords are equal and conversely if two chords of a circle are equal, then their corresponding arcs (minor, major) are congruent.

·       Congruent arcs of a circle subtend equal angles at the centre.

·       The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

·       Angles in the same segment of a circle are equal.

·       Angle in a semicircle is a right angle.

·       If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line segment , the four points lie on a circle.

·       The sum of either pair of opposite angles of a cyclic quadrilateral is 180°.

·       If sum of a pair of opposite angles of a quadrilateral is 180°, the quadrilateral is cyclic.

NUMBER SYSTEMS

NUMBER SYSTEMS

BRIEF SYNOPSIS

·         Numbers that can be expressed in the form  p/q , where q is a non- zero integer and p is any integer are called rational numbers.

·         Numbers that cannot be put in the form p/q , where q≠0, p , q € are called irrational numbers.

·         Every integer is a rational number but a rational number need not be an integer.

·         Every fraction is a rational number but a fraction need not be a rational number.

·         A rational number p/q is said to be in the standard form if q is positive integer and the integer and the integers p and q have no common divisor other than 1.

·         Between two rational numbers x and y , there is a rational number  x + y .

                                                                                                              2

·         We can find as many rational numbers between x and y as we want.

·         If x and y are any two rational numbers, then:

                                 i.            x+ y is a rational number.

                                ii.            x-y is a rational number.

                              iii.            x× y is a rational number.

                              iv.            x÷ y is a rational number (y≠0)

·         Every rational number can be expressed as a decimal.

·         The decimal representation of a rational number is either terminating  or non terminating.

·         If the denominator of a rational number written in standard form contains no prime factors other than 2 or 5 or both, then it can be represented as a terminating decimal.

·         If the denominator of a rational number written in standard form has prime factors other than 2 or 5 or , then it cannot be represented as a terminating decimal.

·          A number is the  terminating decimal form can be converted to one in the rational form by summing the place values of all the digits.

·         A number in the recurring decimal can be converted to one in the rational form by multiplying the decimal by suitable power of 10 and eliminating the decimal point.

·         A  useful result to convert a number in the recurring decimal form to the rational form:

If all the digits on the right side of the decimal part are repeated , then

Given decimal =  Integral part of the decimal number+ Number formed by the digits in decimal part

                                                                                       

                                                                                      Number formed by the same number of 9's as the                  

                                                                                      number of digits in the decimal part

For example :0.7  = 0+ 7  = 7   , 2.49  =2+49 , 5.04 =5+4    etc.

                                       9     9                     99                 99

                                      

·         A number can have decimal representation in one of the following forms:

                     i.          terminating

                   ii.          non- terminating but repeating( recurring)

                  iii.          non-terminating and non -repeating.

The numbers of the type (i) and (ii) are rational numbers where  as of type(iii) are known as irrational.

·         For positive real numbers a and b, the following identities hold:

                                 i.            √ab = √a √b

                                ii.            a

b

                              iii.            (√a+√b) (√a-√b) =a-b

                              iv.            (a+√b)(a-√b)=a²-b

                               v.            (√a+√b)²=a +2√ab+b

·         To rationalise the denominator of        1      , we multiply this by √a-b , where a and b are integers and

                                                               √a +b                                        √a-b                                                                          

·         √a +b≠0 .

·         Let  a> 0 be a real number and p and q be rational numbers .Then

a)    ap

AREAS OF PARALLELOGRAMS AND TRIANGLES

AREAS  OF PARALLELOGRAMS AND TRIANGLES

SOME IMPORTANT POINTS TO REMEMBER

·      Two figures are called congruent, if they have the same shape and the same size.

·      Area of a figure is a number(in some unit) associated with the part of the plane enclosed by the figure with following two properties:

                                                        i.            If  A and B are two congruent figures, then ar (A) = ar (B).

                                                      ii.            If a planar region formed by a figure T is made up of two non-overlapping  planar regions formed by figures P and Q , then ar(T)= ar (P) + ar(Q).

·      Two congruent figures have equal areas but the converse need not be true.

·      Parallelograms on the same base(or equal bases) and between the same parallels are equal in area.

·      Area of a parallelogram is the product of its base and the corresponding altitude.

·      Parallelograms on the same base(or equal bases) and having equal areas lie between the same parallels.

·      If a parallelogram and a triangle are on the same base and between the same parallels, then area of the triangle is half the area of the parallelogram.

·      Triangles on the same base(or equal bases) and between the same parallels are equal in area.

·      Area of a triangle is half the product of its base and the corresponding  altitude.

·      Triangles  on the same base(or equal bases) and having equal areas lie between the same parallels.

·      A median of a triangle divides it two triangles equal areas.

QUADRILATERALS

QUADRILATERALS

SOME IMPORTANT POINTS TO REMEMBER

·      "quadri----- means four" and " lateral ----- means sides".

·      Quadrilateral is a closed figure bounded by four sides such that no two line segments cross each other.

·      The sum of the angles of a quadrilateral is 360°.

·      A quadrilateral having exactly one pair of parallel sides, is called a trapezium.

·      A quadrilateral in which both pairs of opposite sides are parallel, is called a parallelogram.

·      A rectangle is a parallelogram whose angles are right angles.

·      A rhombus is a parallelogram, whose all sides are equal.

·      A square is a parallelogram whose all angles are right angles and all sides are equal.

·      A quadrilateral is a parallelogram , if

                                                        i.   its opposite sides are equal;

                                                      ii.  its opposite angles are equal;

                                                    iii.  its diagonals bisect each other;

                                                    iv.  it has one pair of opposite sides equal and parallel.

·      In a parallelogram

                                                        i.  Diagonals divide it into congruent triangles;

                                                      ii.  Opposite sides are equal;

                                                    iii.  Opposite angles are equal;

                                                    iv.  The diagonals bisect each other.

·      Diagonals of a rectangle bisect each other and are equal and vice-versa.

·      Diagonals of a rhombus bisect each other at right angles and vice -versa.

·      Diagonals of a square bisect each other at right angles and are equal , and vice- versa.

·      The line - segment joining the mid-points of any two sides of a triangle is parallel to the third side and is half of it.

·      A line through the mid -point of a side of a triangle parallel to another side bisects the  third side.

·      The quadrilateral formed by joining the mid -points of the sides of a quadrilateral, in order, is a parallelogram.