Chapter 1 - Number Systems

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.