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\[\overline{AB}\].

    more...

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 more...

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 more...

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 more...

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 more...

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 more...

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 more...

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 more...

 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 more...

 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 more...