When an object or figure to be divided into two equal halves, such that the part coincides with the other, is called symmetrical. If a line divides a given figures into two halves, the line is called line of symmetry or axis of symmetry.
Point– a point determines a location, its has neither length nor thickness. Line– a line is a collection of points which can be extended indefinitely on both the sides it has neither breadth nor thickness that has only length(A line consists of infinite point)
Properties – only one line can be drawn through two points a line has infinitely many points on it. Infinite lines can be drawn through a given point. Plane: Every solid has a surface which is flat or curved. It may be smooth or rough. In geometry we take totally flat or curved surface a plane is a flat surface it has length and width but no height.
Properties of points & lines in a plane. 1. Any two points on the same plane, can be connected with one & only one line passing through them. 2. Two planes intersect in a line. 3. Intersecting lines- two line that meet at a point are called intersecting line.
Parallel line: the lines which never meet even when they are extended are known as parallel lines.
Transversals– a line intersecting two or more given lines in a plane at different point is called a transversal line.
Concurrent lines– three or more lines in a plane are said to be concurrent if all of them pass through same point & this point is called the point of concurence.
Collinear points– Three or more point in a plane are said to be collinear it they all lie on the same line.
Property of divisibility 1. If a number is divisible by another number, it must be divisible by each of the factor or that number. 2. If a number is divisible by each of the two or more than two co prime numbers it must be divisible by their product. 3. If two given number are divisible by a number, then their sum is also divisible by that number. 4. If two given numbers are divisible by a number, than their difference is also divisible by that number.
Cuboid: A solid whose, length, breadth & height are different is called a cuboid. It has a six rectangular plane faces, 12 edges & 8 vertex. Cube: A cuboid whose length ,breadth & height are equal is called a cube. Cylinder: When a rectangle. OABO’ is rotate at its oo’ axis in initial position to thus form a solid is called cylinder . Cone: When a triangle AOV is rotate to its OV axis in initial position to thus form a solid is called cone. Prism: When a traingle raise a required height H thus form a solid is called prism. Cubes and cuboids are also known as square prism and rectangle prism.
Sphere: When a circle is rotate at its AOB axis in initial position to thus form a solid is called sphere. Pyramid:A solid whose base is triangle or square in shape & it’s all vertices are meet a point at height H. Triangular pyramid: A pyramid having triangular base is called triangular pyramid. He is also called a tetrahedeon (four faced sided ) Square pyramid: A pyramid having a square base is called a square pyramid. Continued proportion e.g. 4:3::8:16 Mean proportion e.g. 4:2::8:16 or 82 = 4×16 8 is mean proportion. Third proportion 4:8::8:16 – 16 is called third proportion. Unitary method The method of finding is the value of one unit from the values of given own.
Ratio– The ratio of two a & b is a/b & is denoted a:b where a & b are called the terms of the ratio.; the ist term a is called antecedent & the second term b is called consequence . ratio of two quontities is just a number & has no unit at all the comparison are made only between same kind of quantity.
Comparison of ratio– Express each of the ration in the simplest form of a friction find the LCM of the denominator of the fraction convert the denominator of each fraction equal to LCM and numerators according changes in denominator .
Proportion – The proportion is a statement that two ratio are equal in other words it two ratio is equal we say that they are in proportion & use symbol: (to equal the two ratio) e.g. a:b :: c:d
put 1 in the denominator under the decimal point & annexure with it as many zero as is the number of digits after the decimal points now remove the decimal point & reduce the fraction of its lowest term.
3) i) annexing zero to the extreme right of a decimal fraction does not change its value. Thus 0.80= 0.800 etc.
ii) if numerator & denominator of a fraction contain the same number of decimal places then we move the decimal sign. Thus,
Operation on decimal fraction
) Addition & Substraction– given number are so placed under each other that the decimal points lie in the correspondent column and now be added or subtracted in the usual way.
ii) Multiplication of a decimal fraction by a power of 10 Shift the decimal point to the right by as many phase as is the power of 10
4.31 10 43.1/10 4.6987 1000 = 4698.7/1000
iii) Multiplication of decimal fraction:-
multiply the given numbers considering them without the decimal point.
Now in the product, the decimal point is marked of to obtain an any places of decimal as is the sum of the number of decimal places in the given number.
Example (sum of decimal places 1+1+1+3=6 no.)
iv) Dividing a decimal fraction by a converting number–
divide the given number without considering the decimal point by the given counting number now in the quotient put the decimal point to give as many places of decimal as there are in the dividend.
v) Dividing a decimal fraction by a decimal fraction
Comparison of fraction–
fractions are to be arranged in ascending or dis cending order of magnitude, then convert each one of the given fraction in the decimal form & arrange them accordingly.
Here So >
vi) Recurring decimal–
if in a decimal fraction, a figure or a set of figures is repeated continuously then such a number is called a recurring decimal thus. ,
Pure recurring decimal–
A decimal fraction in which all the figure. After the decimal point are repeated is called a pure recurring decimal.
Converting a pure recurring decimal into vulgar fraction
Write the repeated figures only one in numerator & take as many mix in the denominators as is the number of repeating figure.
Mixed recurring decimal–
A decimal fraction in which some figure do not repeat & some of them are repeated, is called mixed recurring decimal.
Converting a mixed recurring decimal into vulgar fraction
In the numerator take the difference between the number formed by all the digit after decimal point . (taking repeated digit only once) & that formed by digit which are not repeated.
In the denominator take the number formed by as many as there are repeating digit followed by as many zeros as is the number of non repeating digits. Thus,
A fraction is a number representing a part of the whole e.g. ¼ (1 is the numerator & 4 is the denominator ) is showing whole 4 than ¼ is the 1/4th part. Like fraction– fraction having the same denominator are called like fraction. Unlike fraction– fractions having different denominator are called unlike fraction. Proper fraction– a fraction whose numerator is less than the denominator is called proper fraction.e.g. 2/3 Improper fraction– a fraction whose numerator is greater than the denominator is called improper fraction. Unit fraction– a fraction having numerator is 1 called a unit fraction ½, ¼,1/7 etc. Mixed fraction– a fraction which is a combination of a whole number and a proper fraction is called a mixed fraction. E.g. 2 1/(3 ) Equivalent fraction– ½ = all these fractions being equal in value are called equivalent fractions.
Comparing Fractions 1. We find the LCM of their denominators. 2. We make the denominator of each fraction equal to the LCM & convert them into like fraction. 3. Then compare the numerator of fraction. and = and here we can see that
Addition & subtraction of fraction– same to comparing procedure . Fraction– a number of the form a/b where a & b are whole number & is called a fraction, a is called numerator b is the denominator a fraction is the number representing a part of the whole. Sum/difference of fraction = sum/difference of their numerator / common denominator
H.C.F- The highest common factor of two or more given number is the highest (or greatest) of their common factor. e.g. 24 = 1,2,3,4,6,8,12,24 36 = 1,2,3,4,6,9,12,36 H.C.F = 6
Euclid’s Algorithm determining the HCF of 18 & 30.
Lowest /least common multiple- the lowest common multiple of two numbers is smallest natural number which is a multiple of both the number. Example LCM of 4 an d 6 Multiple of 4 are 8 12 16 20 Multiple of 6 are 12 18 24 So LCM of 4 and 6 are 12
The product of the H.C.F & L.C.M of two numbers is equal to the product of the number. Example: Given number 16 and 24. their and then
Absolute value of integer |-5| = 5, but -|5| = 5, |+5| = 5
Quadrilaterals- a simple closed figure surrounded by four line segment is called a quadrilaterals. It has four sides , four angles, four vertices and two diagonals. The sum of all angles are . Adjacent side- having a common end point. e.g PQ & QR Opposite side- do not have any common point. eg. PQ, RS
Similarly the adjacent angles opposite angle. The sum of the angle of a quadrilateral is .
Type of quadrilaterals: Rectangle– a quadrilateral in which opposite side are parallel & equal & each angle is called a rectangle.
Square– a rectangle in which all sides are equal opposite side are, parallel and each angle is , called a square.
Parallelogram– a quadrilateral opposite sides are parallel & opposite angles are equal is called parallelogram.
Rhombus– a quadrilateral which all side are equal , opposite side are parallel and opposite angles are equal is called rhombus. Trapezium– a quadrilateral having one opposite side is parallel is called trapezium. When if its non parallel sides are equal is called isosceles trapezium. a quadrilateral is called if their adjacent sides are equal and opposite sides are unequal.
The figure formed by joining three non collinear points by line segment is called a triangle. It has three sides, 3 vertex and 3 angle. The sum of interior angles of triangle is . Exterior angle of triangle– The exterior angle of a triangle is equal to sum of opposite interior angles. The sum of two side of a triangle is greater than third side.
Here exterior angle of A=Interior angle of B+Interior angle of C Between two parallel line and a common base, formed all triangle is equal in area.
In above diagram
Perimeter of triangle– the sum of the length of all side of a triangle is called perimeter of triangle.
Altitude of a triangle– the perpendicular drawn from the vertex of a triangle to the opposite side is called an altitude . all altitude meet at a point. Median of a triangle– the line joining any vertex of a triangle to the mid-point of its opposite side is called the median of the triangle. That medians meet at a point. Congruent triangles– two triangles are said to be congruent if every angle of one is equal to corresponding angle of the other and every side of one is equal to the corr esponding side of the other. Scalene triangle – A triangle whose no side are equal.
Isosceles triangle – A triangle whose any two side are equal to each other is called an isosceles.
The opposite angle of equal side is equal
Equilateral triangle – A triangle having all side equal to one another is called an equilateral triangle.
In equilateral triangle all angles are equal if then
Acute angle triangle– A triangle in which all the angles are acute angles . Here each angle is less that Right angle triangle– A triangle whose one angle is is called an right angle triangle. Obtuse angle triangle– A triangle whose one angle is obtuse more than is called an obtuse angled triangle. Area of triangle 1. 2.