Current Affairs 7th Class

Lines and Angles

Point: A point is a geometrical representation of a location. It is represented by a dot.

Line: A geometrical line is a set of points that extends endlessly in both the directions i.e., a line has no end points. A line AB is represented as.\[\overleftrightarrow{AB}\]

Line segment: A line segment is a part of a line. A line segment has two end points. A line segment AB is represented as\[\overline{AB}\].

Ray: A ray is a part of the line which has one end point (namely its starting point).

A ray OP is denoted as\[\overrightarrow{OP}\].

Angle: An angle is the union of two rays with a common initial point.

The symbol of angle is\[\angle \]. An angle is measured in degrees\[\left( {}^\circ \right)\]. The angle formed by the two rays \[\overrightarrow{AB\,}\,and\,\overrightarrow{AC}\]is denoted by \[\angle \]BAC or \[\angle \]CAB

The two rays \[\overrightarrow{AB\,}\,and\,\overrightarrow{AC}\] are called the arms and the common initial point 'A' is called the vertex of the angle ABC.

Types of Angles:

(i) Right angle: An angle whose measure is equal to \[{{90}^{o}}\]is called a right angle.

(ii) Acute angle: An angle whose measure is less than 90° is called an acute angle.

(iii) Obtuse angle: An angle whose measure is greater than 90° but less than 180° is called an obtuse angle. more...

Catgeory: 7th Class

Triangles

A triangle is a simple closed figure bounded by three line segments. It has three vertices three sides and three angles. The three sides and three angles of a triangle are called its six elements. It is denoted by the symbol A.

In\[\Delta \]ABC. Sides: \[\overline{AB}\,,\overline{BC}\,and\,\overline{CA}\]; Angles: \[\angle \]BAC,\[\angle \]ABC and \[\angle \]BCA ; Vertices: A, Band C

A triangle is said to be

(a) an acute angled triangle, if each one of its (b) a right angled triangle, if any one of its angles measures \[{{90}^{o}}\]. (c) an obtuse angled triangle, if any one of its angles measures more than \[{{90}^{o}}\] Note: A triangles cannot have more than one right angle. A triangles cannot have more than one abuts angle. In a right triangle, the sum of the acute angles is \[\mathbf{9}{{\mathbf{0}}^{\mathbf{o}}}\]

Angle sum property: The sum of the angles of a triangle is \[{{180}^{o}}\].
Properties of sides:

(i) The sum of any two sides of a triangle is greater than the third side. (ii) The difference of any two sides is less than the third side. (iii) Property of exterior angles: If a side of a triangle is produced, the exterior angle so formed is equal to the sum of interior opposite angles.

e.g., Exterior angle, \[{{x}^{o}}=\angle A+\angle B={{70}^{o}}+{{40}^{o}}=\,{{110}^{o}}\]

A triangle is said to be

(a) an equilateral triangle, if all of its sides are equal. (b) an isosceles triangle, if any two of its sides are equal. (c) a scalene triangle, if all of its sides are of different lengths.

The medians of a triangle are the line segments joining the vertices of the triangle to the midpoints of the opposite sides.

Here AD, BE and CF are medians of \[\Delta \]ABC.

The medians of a triangle are concurrent.
The centroid of a triangle is the point of concurrence of its medians. The centroid is denoted by G.
Triangle divides the medians in the ratio 2:1.
The medians of an equilateral triangle are equal.
The medians to the equal sides of an isosceles triangle are equal.
The centroid of a triangle always lies in the interior of the triangle.

Altitudes of triangle are the perpendiculars more...

Catgeory: 7th Class

Congruence of Triangles

Two figures having exactly the same shape and size are said to be congruent.
Two triangles are said to be congruent, if pairs of corresponding sides and corresponding angles are equal.

Note: The symbol\[\cong \]is used for ‘is Congruent to’ relation.

Two line segments are congruent, if they have the same length. \[\overline{AB} = \overline{CD}\]is read as line segment \[\overline{AB}\]is congruent to the line segment \[\overline{CD}\]

Two angles are congruent, if they have the same measure. “\[\angle \]A is congruent to \[\angle \]B” is written symbolically as \[\angle \]A = \[\angle \]B or \[\angle \]A = \[\angle \]B.

S.S. congruence condition: If the three sides of a triangle are equal to the three corresponding sides of another triangle, then the two triangles are congruent.

e.g.

In the given figure, \[\Delta \]ABC =\[\Delta \]DEF by S.S.S. congruence condition.

A.S. congruence condition: If two sides and the included angle of a triangle are respectively equal to the two corresponding sides and the included angle of another triangle, then the two triangles are congruent.

e.g.

in the given figure, AABC '= ADEF by S.A.S. congruence condition.

S.A. congruence condition: If two angles and an included side of one triangle are re- spectively equal to the two corresponding angles and the corresponding included side of another triangle, then the two triangles are congruent.

e.g.

In the given figure,\[\Delta \]ABC=\[\Delta \]DEF by A.S.A. congruence condition.

H.S. congruence condition: If the hypotenuse and a side of a right angled triangle are equal to the hypotenuse and the corresponding side of another right-angled triangle then the two triangles are congruent.

e.g.

In the given figure,\[\Delta \]ABC\[\cong \]\[\Delta \]DEF by R.H.S. more...

Catgeory: 7th Class

Comparing Quantities

Ratio is a method of comparing two quantities of the same kind by division.
The symbol used to write a ratio is'.-'and is read as \[';'\]is to'.
A ratio can be expressed as a fraction.
A ratio is always expressed in its simplest form.
A ratio does not have any unit, it is only a numerical value.
A ratio consists of two terms. The first term is called the antecedent and second term is called the consequent.

A ratio can be written in its simplest form by dividing the antecedent and the consequent by their H.C.F.

The antecedent and the consequent of a ratio cannot be interchanged.
To express two terms in a ratio, they should be in the same units of measurement.
When two ratios are equal they are said to be in proportion, the symbol for proportion is \[::\] and is read as 'as to'.
The two terms in the middle of a proportion are called means and the first and the last terms are called extremes.
If two ratios are to be equal or to be in proportion, their product of means should be equal to the product of extremes.
lf a:b :: c:d then the statement ad = be holds good.
lf a:b and b: c are in proportion such that \[{{\operatorname{b}}^{2}}=ac\]hen b is called the mean proportional of a:b and b:c.
Multiplying or dividing the terms of the ratio by the same number gives equivalent ratios.

Unitary method:
To find the value of many quantities when the value of one is given, the operation is multiplication (x).To find out value of one when the value of many is given. The operation is division (-).To find out value of many when the value of many is given, unitary method can be used.

Another way of comparing quantities is percentage. The word percent means per hundred. Thus 12% means 12 parts out of 100 parts.
Fractions can be converted into percentages and vice-versa.

e.g., (i)\[\frac{2}{5}=\frac{2}{5}\times 100%=40%\] (ii)\[25%=\frac{25}{100}=\frac{1}{4}\]

Decimals can be converted into percentages and vice-versa.
g. (i) \[0.36=0.36\times 100%=36%\]

(ii)\[43%=\frac{43}{100}=0.43\]

If a number is increased by a% and then decreased by a% or is decreased by a% and then increased by a%, then the original number decreases by \[\frac{{{a}^{2}}}{100}\]%.
A number can be split into two parts such that one part is P% of the other. Then the two parts are

\[\frac{100}{100+2}\times \,number\,and\,\frac{p}{100+P}\times number\]

If the circumference of a circle is increased (or) decreased by P% then the radius of a circle in- increases (or) decreases by P%,
Gain more...

Catgeory: 7th Class

Rational Numbers

Natural numbers (N): 1, 2, 3, 4 ... etc., are called natural numbers.
Whole numbers (W): 0, 1, 2, 3...... etc., are called whole numbers.
Integers (Z): ....... -3, -2, -1, 0, 1, 2, 3 …… etc., are called integers, denoted by I or Z.

1, 2, 3, 4, etc., are called positive integers denoted by 7. \[{{Z}^{+}}\]. -1, -2, - 3, - 4,...... etc., are called negative integers denoted by Z- Note: 0 is neither positive nor negative.

Fractions:

The numbers of the form \[\frac{x}{y}\], where x and y are natural numbers, are known as fractions. e.g.\[\frac{3}{5},\frac{2}{1},\frac{1}{125},\]......... etc.

Rational numbers (Q):

A number of the form \[\frac{p}{q}\](q\[\ne \]O), where p and q are integers is called a rational number. e.g.,\[\frac{-3}{17},\frac{5}{-19},\frac{10}{1},\frac{-11}{-23},\]…. etc. Note: 0 is a rational number, since 0=\[\frac{\mathbf{0}}{\mathbf{1}}\].

A rational number r- is positive if p and q are either both positive and both negative.

e.g., \[\frac{3}{5},\frac{-2}{-7}\]

A rational number \[\frac{p}{q}\]is negative if either of p and q is positive and the other is negative.

e.g., \[\frac{-5}{3},\frac{7}{-23}\] Note: 0 is neither a positive nor a negative rational number.

Representation of Rational numbers on a number line:

We can mark rational numbers on a number line just as we do integers. The negative rational numbers are marked to the left of 0 and the positive rational numbers are marked to the right of 0. Thus,,\[\frac{1}{3}and-\frac{1}{3}\]would be at an equal distance from 0 but on its either side. Similarly, other rational numbers with different denominators can also be represented on the number line. Thus, in general, any rational number is either of the following two types. (i)\[\frac{m}{n}\] where m < n (ii)\[\frac{m}{n}\] where m > n e.g., 4, 6 \[\frac{1}{2},\frac{3}{4},\frac{5}{6}\]etc., e.g.,\[\frac{7}{6},\frac{3}{2},\frac{15}{6}\] etc. Representation of\[\frac{\mathbf{m}}{\mathbf{n}}\] on the number line where m < n: The rational number - (5 < 6) is represented on the number line as shown.

Representation of \[\frac{\mathbf{m}}{\mathbf{n}}\]on the number line where m > n:

Consider the rational number \[\frac{17}{5}\]. First convert the rational number\[\frac{17}{5}\]into a mixed fraction and then mark it on the number line i.e.

Standard form of a rational number:

A rational number'\[\frac{p}{q}\]is more...

Catgeory: 7th Class

Practical Geometry

A ruler and compasses are used for constructions.
Given a line \[l\] and a point not on it, a line parallel to \[l\] can be drawn using the idea of 'equal alternate angles' or 'equal corresponding angles'.
Three independent measurements are required to construct a triangle.
A rough sketch is drawn with the given measurements before actually constructing the triangle.
The sum of lengths of any two sides of a triangle is greater than its third side.
The difference of lengths of any two sides of a triangle is lesser than its third side.
The sum of angles in a triangle is\[{{180}^{o}}\].
The exterior angle of a triangle is equal in measure to the sum of interior opposite angles.

The following cases of congruence of triangles are used to construct a triangle.

(i) S.S.S: A triangle can be drawn given the lengths of its three sides. (ii) S.A.S: A triangle can be drawn given the lengths of any two sides and the measure of the angle between them. (iii) A.S.A: A triangle can be drawn given the measures of two angles and the length of the side included between them. (iv) R.H.S: A triangle can be drawn given the length of hypotenuse of a right angled triangle and the length of one of its legs.

A triangle is said to be,

(a) an equilateral triangle, if all of its sides are equal.(b) an isosceles triangle, if any two of its sides are equal. (c) a scalene triangle, if all of its sides are of different lengths.

A triangle is said to be,

(a) an acute angled triangle, if each one of its angles measures less than \[{{90}^{o}}\]. (b) a right angled triangle, if any one of its angles measures\[{{90}^{o}}\]. (c) an obtuse angled triangle, if any one of its angles measures more than 90°.

Pythagoras' theorem: In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the remaining two sides.

\[\operatorname{Here},A{{C}^{2}}=A{{B}^{2}}+B{{C}^{2}}\]

Catgeory: 7th Class

Perimeter and Area

Perimeter is the distance around a closed figure.
Area is the part of plane occupied by the closed figure.

(a) Perimeter of a square= 4\[\times \]side units (b) Perimeter of a rectangle = 2\[\times \](length + breadth) units (c) Area of a square = side \[\times \] side sq. units (d) Area of a rectangle = length \[\times \] breadth sq. units

Area of a parallelogram = base \[\times \] height sq. units
Area of a triangle =\[\frac{1}{2}\](area of the parallelogram generated from it)

= \[\frac{1}{2}\] \[\times \]base \[\times \] height sq. units

Area of a trapezium =\[\frac{1}{2}\](a + b) h sq. units, where \['a'\]and \['b'\] are lengths of parallel sides and 'h' is the height.

A circle is a closed curve in a plane drawn in such a way that every point on it is at a constant distance (r units) from a fixed point 0 inside it.

The fixed point 0 is called the centre of the circle and the constant distance r is called the radius of the circle.

Circumference of a Circle: The perimeter of a circle is called its circumference.
Circumference =\[2\pi r\] =\[\pi \]d, where r = radius and d = diameter.

Here \[\pi \](Pi) is a constant, equal to 3.14 approximately.

Area of a Circle: Area of a circle with radius r units is equal to \[\pi {{r}^{2}}\] sq units.
Area of a Ring:

The region enclosed between two concentric circles of different radii is called a ring.

Area of path formed between two concentric circular regions \[=\left( \pi {{R}^{2}}-\pi {{r}^{2}} \right)\] sq. units \[=\pi \left( {{R}^{2}}-{{r}^{2}} \right)\]square units \[=\pi \left( R+r \right)\left( R-r \right)\]square units

Catgeory: 7th Class

Algebraic Expressions

Algebra: It is a branch of mathematics in which we use literal numbers and statements symbolically. Literal numbers can be positive or negative. They are variables.

Variable: A symbol which takes various values is known as a variable. Normally it is denoted by x, y, z etc.

Constant: A symbol having a fixed numerical value is called a constant. Sometimes, 'c', 'k', etc., are used as symbols to denote a constant.

Coefficient: In a term of an algebraic expression any of the factors with the sign of the term is called the coefficient of the product of the other factors in that term. Sometimes, symbols like a, b,\[l\], m etc., are used to denote the coefficients. Coefficients that are numbers are called numerical coefficients.

Algebraic expression: A combination of constants and variables connected by some or all of the four fundamental operations +, -, x and - is called an algebraic expression.

e.g., - 5p + 12 is an algebraic expression. Here -5 is the coefficient of the variable 'p' and 12 is the constant.

Terms of an algebraic expression: The different parts of the algebraic expression separated by the sign + or -, are called the terms of the expression.

e.g., 3x - 5 + 4xy is an algebraic expression containing 3 terms -3x, -5 and 4xy.

Like and unlike terms: In a given algebraic expression, the terms having the same literal factors are called like or similar terms, otherwise they are called unlike terms.

e.g., 3xy and -4xy are like terms while 6xy and -4x are unlike terms.

Factors: Each term of an algebraic expression consists of a product of constants and variables.

A constant factor is called a numerical factor, while a variable factor is known as a literal factor.

Various types of algebraic expressions:

(i) Monomial: An algebraic expression which contains only one term, is called a monomial, Thus,\[5x,\text{ }2xy,-3{{a}^{2}}b,-7,\]etc., are all monomials. (ii) Binomial: An algebraic expression containing two terms is called a binomial. Thus,\[\left( 2a+3b \right),\left( 8-3x \right),\left( {{x}^{2}}-4x{{y}^{2}} \right),\]etc., are all binomials. (iii) Trinomial: An algebraic expression containing three terms is called a trinomial.

Thus, \[\left( a+2b+5c \right),\left( x+2y-3z \right),({{x}^{3}}-{{y}^{3}}-{{z}^{3}}),\]etc., are all trinomials.

(iv) Polynomial: An expression containing two or more terms is called a polynomial.

Addition of Algebraic Expressions: While adding algebraic expressions, we collect the like terms and add them. The sum of several like terms is another like term whose coefficient is the sum of the coefficients of those like terms. The like terms are added and the unlike terms are left as they are.

more...

Catgeory: 7th Class

Exponents and Powers

Exponential form is the short form of repeated multiplication. A number written in exponential form contains a base and an exponent.

\[{{10}^{5}}\]is the exponential form of 1,00,000, since 1,00,000 =10\[\times \]10\[\times \]10\[\times \]10\[\times \]10. In \[{{10}^{5}},\,10\] is the base and 5 is the exponent or index or power.

Base denotes the number to be multiplied and the power denotes the number of times the base is to be multiplied.

\[a\times a={{a}^{2}}\](read as 'a squared' or 'a raised to the power 2') \[a\times a\times a={{a}^{3}}\](read as 'a cubed' or 'a raised to the power 3') \[a\times a\times a\times a={{a}^{4}}\] (read as 'a raised to the power 4' or \[{{4}^{th}}\] power of a) ……………………………………………………. \[a\times a\times a\,....\] (n factors) \[={{a}^{n}}\](read as 'a raised to the power n' or \[{{\operatorname{n}}^{th}}\]power of a)

(i) When a negative number is raised to an even power the value is always positive.

e.g.,\[{{\left( -5 \right)}^{4}}=\left( -5 \right)\times \left( -5 \right)\times \left( -5 \right)\times \left( -5 \right)=+\,625\]

When a negative number is raised to an odd power, the value is always negative.

e.g., \[{{\left( -3 \right)}^{5}}=\left( -3 \right)\times \left( -3 \right)\times \left( -3 \right)x\left( -3 \right)x\left( -3 \right)=\left( -\,243 \right)\] Note: (a) \[{{\left( -1 \right)}^{oddnumber}}=-1\] (b)\[{{\left( -1 \right)}^{oddnumber}}=+1\]

Laws of Exponents:

For any non-zero integers 'a' and V and whole numbers 'm' and 'n', (i) \[a\times a\times a\times ......\times a\](m factors)\[={{a}^{m}}\] (ii) \[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\] (iii) \[\frac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m+n}},\operatorname{if}\,\,m>n\] \[=1,\text{ }if\text{ }m=n\] \[=\frac{1}{{{a}^{n-m}}}\,\operatorname{if}\,m<n\] (iv) \[{{\left( {{a}^{m}} \right)}^{n}}{{a}^{mn}}\] (v) \[{{\left( ab \right)}^{m}}={{a}^{m}}{{b}^{m}}\] (vi) \[{{\left( \frac{a}{b} \right)}^{m}}=\frac{{{a}^{m}}}{{{b}^{m}}}\] (vii)\[{{a}^{o}}=1\]

Any number can be expressed as a decimal number between 1.0 and 10.0 including 1.0 multiplied by a power of 10. Such a form of a number is called its standard form.

Catgeory: 7th Class

Symmetry

Linear symmetry: If a line divides a given figure into two coinciding parts, we say that the figure is symmetrical about the line and the line is called the axis of symmetry or line of symmetry.

A line of symmetry is also called a mirror line.
A figure may have no line of symmetry, only one line of symmetry, two lines of symmetry or multiple lines of symmetry.
Regular polygons have equal sides and equal angles. They have multiple lines of symmetry.
Each regular polygon has as many lines of symmetry as its sides.
A scalene triangle has no line of symmetry.
A parallelogram has no line of symmetry.
A line segment is symmetrical about its perpendicular bisector.
An angle with equal arms has one line of \[\leftrightarrow \]symmetry.
An isosceles triangle has one line of symmetry.
An isosceles trapezium has one line of symmetry.
A semicircle has one line of symmetry.
A kite has one line of symmetry.
A rectangle has two lines of symmetry.
A rhombus has two lines of symmetry.
An equilateral triangle has three lines of symmetry
A square has four lines of symmetry.
A circle has an infinite number of lines of symmetry.
In English alphabet, the letters A, B, C, D, E, K, M, T, U, V, W and Y have one line of symmetry and the letters H, I, X have two lines of symmetry
In English alphabet, the letters F, GJ, L, N, P, Q, R, S and Z have no line of symmetry The letter 0 has many lines of symmetry.

The line symmetry is closely related to mirror reflection. When dealing with mirror reflection, we have to take into account the left\[\leftrightarrow \] right changes in orientation.
Point symmetry: A figure is said to be symmetric about a point 0, called the centre of symmetry, if corresponding to each point P on the figure, there exists a point P' on the other side of the centre, which is exactly opposite to the point P and lies on the figure.

Note: A figure that possesses a possesses a point symmetry, regains its original shape even after beging rotated through \[\mathbf{18}{{\mathbf{0}}^{\mathbf{o}}}\]