## TS 7th Class Maths Concept

##### Studying maths in VII class successfully meaning that children take responsibility for their learning and learn to apply the concepts to solve problems.

##### This notes is designed by the ‘BasicinMaths’ team. These notes to do help students fall in love with mathematics and overcome fear.

** 1. INTEGERS**

** Natural numbers:** All the counting numbers starting from 1 are called Natural numbers.

1, 2, 3… Etc.

** Whole numbers:** Whole numbers are the collection of natural numbers including zero.

0, 1, 2, 3 …

** Integers: **integers are the collection of whole numbers and negative numbers.

….,-3, -2, -1, 0, 1, 2, 3,…..

** ****Integers on a number lin**e:

** Operations on integers:**

** addition of integers:**

3 + 4 = 7

-2 + 4 = 2

** Subtraction of integers on a number line:-**

6 – 3 = 3

** Multiplication of integers on a number line:-**

2 × 3 ( 2 times of 3) = 6

3 × (- 4 ) ( 3 times of -4) = -12

** Multiplication of two negative integers:**

- To multiply two negative integers, first, we multiply them as whole numbers and put plus sign before the result.
- Multiplication of two negative integers is always negative.

Ex:- -3 × -2 = 6, -10 × -2 = 20 and so on.

** Multiplication of more than two negative integers:**

• If we multiply three negative integers, then the result will be a negative integer.

Ex:- -3 × -4 × -5 = -60, -1× -7 × -4 = -28 and so on.

• If we multiply four negative integers, then the result will be a positive integer.

Ex:- -3 × -4 × -5 × -2 = 120, -1× -7 × -4 × -2 = 56 and so on.

** Note:-**

1. If the no. of negative integers is even, then the result will be positive.

2. If the no. of negative integers is odd, then the result will be negative.

** Division of integers:**

- The division is the inverse of multiplication.
- When we divide negative integer by a positive integer or positive integer by negative integer, we divide them as whole numbers then put negative signs for the quotient.

Ex:- -3 ÷ 1 = 3, 4 ÷ -2 = -2 and so on.

- When we divide a negative integer by a negative integer, we get a positive number as the quotient.

Ex:- -3 ÷ -1 = 3, -4 ÷ -2 = 2 and so on.

** Properties of integers:**

** 1.Closure property:-**

** 2.commutative property:-**

**3.associative property:-**

**Additive identity:-**

1 + 0 = 0 + 1 = 1, 10 + 0 = 0 + 10 = 10

•For any integer ‘a’, a + 0 = 0 + a

•0 is the additive identity.

**Additive inverse:- **

2 + (-2) = (-2) + 2 = 0, 5 + (-5) = (-5) + 5 = 0

•For any integer ‘a’, a+ (-a) = (-a) + a = 0

•Additive inverse of a = -a and additive inverse of (-a) = a

**Multiplicative identity:-**

2 × 1 = 1 × 2 = 2, 5 × 1 = 1 × 5 = 5

•For any integer ‘a’, a × 1 = 1 × a = a

•1 is the multiplicative identity.

**multiplicative inverse:-**

For any integer ‘a’, 1/a × a = a × 1/a = 1

- multiplicative inverse of a = 1/a
- Multiplicative inverse of 1/a = a.

**distributive property:- **

For any three integers a, b and c, a × (b + c) = (a × b) + (a × c).

3 × (2 + 4) = 18

(3 × 2) + (3 × 4) = 6 + 12 = 18

∴ 3 × (2 + 4) = (3 × 2) + (3 × 4).

**2. FRACTIONS, DECIMALS AND RATIONAL NUMBERS**

**Fraction: A fraction** is a number that represents a part of the whole. A group of objects is divided into equal parts, then each part is called a fraction.

** The proper and improper fractions:**

In a **proper fraction,** the numerator is less than the denominator.

Ex: – 1/5, 2/3, and so on.

In an **improper fraction,** the numerator is greater than the denominator.

Ex: – 5/2,11/5 and so on.

**Comparing fractions:**

**Like fractions: – **We have to compare the like fractions with the numerator only because the like fractions have the same denominator. The fraction with the greater numerator is greater and the fraction with the smaller numerator is smaller.

**Unlike fractions: – **

**With the same numerator: **For comparing unlike fractions, we have to compare denominators when the numerator is the same. The fraction with a greater denominator is smaller and the fraction with the smaller denominator is smaller.

**Note: – **To find the equivalent fractions of both the fractions with the same denominator, we have to take the LCM of their denominators.

**Addition of fractions:**

∗ **Like Fractions:**

∗ Unlike** fractions:**

**Subtraction of fractions:**

**∗ Like fractions:**

**Unlike fractions: –** Firs**t,** we have to find the equivalent fraction of given fractions and then subtract them as like fractions

**Multiplication of fractions:**

**Multiplication of fraction by a whole number: –**

Multiplication of numbers means adding repeatedly.

• To multiply a whole number with a proper or improper fraction, we multiply the whole number with the numerator of the fraction, keeping the denominator the same.

2.**Multiplication of fraction with a fraction: –**

multiplication of two fractions =

**Division of fractions:**

⇒ 6 one-thirds in two wholes

**Reciprocal of fraction: **reciprocal of a fraction

**Note:**

- dividing by a fraction is equal to multiplying the number by its reciprocal.
- For dividing a number by mixed fraction, first, convert the mixed fraction into an improper fraction and then solve it.

**1.Division of whole number by a fraction: –**

**2.Division of a fraction by another fraction: –**

** ** **Decimal number or fractional decimal: **

In a decimal number, a dot(.) or a decimal point separates the whole part of the number from the fractional part.

The part right side of the decimal point is called the **decimal part** of the number as it represents a part of 1. The part left to the decimal point is called the **integral part** of the number.

Note: –

- while adding or subtracting decimal numbers, the digits in the same places must be added or subtracted.
- While writing the numbers one below the other, the decimal points must become one below the other. Decimal places made equal by placing zeroes on the right side of the decimal numbers.

**Comparison of decimal numbers:**

while comparing decimal numbers, first we compare the integral parts. If the integral parts are the same, then compare the decimal part.

Ex: – which is bigger: 13.5 or 14.5

Ans: 14.5

Which is bigger: 13.53 or 13. 25

Ans: 13.53

**Multiplication of decimal numbers:**

For example, we multiply 0.1 × 0.1

**Multiplication of decimal numbers by 10, 100, and 1000: –**

Here, we notice that the decimal point in the product shifts to the right side by as many zeroes as in 10, 100, and 1000.

**Division of decimal number**:

**Division of decimal number by 10,100 and 1000: –**

Here, we notice that the decimal point in the product shifts to the left side by as many zeroes as in 10, 100, and 1000.

**Rational numbers:**

** **The numbers which are written in the form of p/q, where p, q are integers, and q ≠ 0, are called rational numbers.

Rational numbers are a bigger collection of integers, negative fractional numbers, positive fractional numbers.

Ex: – 1, 2, -1/2, 0 etc.

**3. SIMPLE EQUATIONS**

**Equation: **Equation is the condition on a variable. It says that two expressions are equal.

- An equation has two sides LHS and RHS, on both sides of equality of sign.
- One of the expressions of the equation is must have a variable.
- If we interchange the expressions from LHS to RHS, the equation remains the same

Ex: – x + 2 = 5; 2 = x + 3

**Balanced equation:**

In an equation, if LHS =RHS, then that equation is balanced.

If the same number is added or subtracted on both sides of the balanced equation, the equation remains will the same.

Ex: 8 + 3 = 11

If add 2 on both sides ⇒ LHS = 8 + 3 + 2 = 13

RHS = 11 + 2 = 13

∴ LHS = RHS

8 +3 = 11 if subtract 2 on both sides

LHS = 8 + 3 – 2 = 9

RHS = 11 – 2 = 9

∴ LHS = RHS

**Using algebraic equations in solving day to day problems:**

- Read the problem carefully.
- Denote the unknown or he quantity to be found with some letters such as x, y, z …etc.
- Write the problem in the form of an algebraic equation by making a relation among the quantities.
- Solve the equation.
- Check the solution

**4. LINES AND ANGLES**

**Complimentary angles: **When the sum of the angles is 90^{0}, the angles are called complementary angles.

**Ex:** 30^{0}, 60^{0}; 20^{0}, 70^{0} and soon.**Supplementary angles: **When the sum of the angles is 180^{0}, the angles are called Supplementary angles.

**Ex:** 120^{0}, 60^{0}; 110^{0}, 70^{0} and soon.

**Adjacent angles: **The angle having a common arm and a common vertex are called Adjacent angles.

⇒ ∠AOC and ∠BOC adjacent angles.

**Vertically opposite angle: **If two lines are intersecting at a point, then the angles that are formed opposite to each other at that point are called vertically opposite angles.

**Transversal: **A line that intersects two or more lines at distinct points is called a transversal.

**Angles made by a transversal:**

**Corresponding angles: –**

Two angles which are lies the same side of the transversal and one interior and another one exterior are called corresponding angles.

∠1, ∠5; ∠2, ∠6; ∠3, ∠7 and ∠4, ∠8

**Alternate angles: –**

Two angles which are lies opposite side of the transversal and both interior or exterior are called corresponding angles.

∠1, ∠7; ∠2, ∠8 are exterior alternate angles

∠3, ∠5; ∠4, ∠6 are interior alternate angles.

∠3, ∠6; ∠4, ∠5 interior angles same side of the transversal.

**Transversal on parallel lines:**

• If pair of parallel lines are intersected by a transversal then the angles each pair of **corresponding angles** are equal

⇒ ∠1, =∠5; ∠2= ∠6; ∠3= ∠7 and ∠4= ∠8

•If pair of parallel lines are intersected by a transversal then the angles each pair of **interior alternate angles** are equal.

∠3= ∠5; ∠4= ∠6

•If pair of parallel lines are intersected by a transversal then the angles each pair of **exterior alternate angles** are equal.

∠1= ∠7; ∠2= ∠8

•If pair of parallel lines are intersected by a transversal then the angles each pair of **interior angles in the same side if the transversal **are supplementary.

∠3+∠6= 180^{0}; ∠4+ ∠5 = 180^{0}

**Note:**

- If a transversal intersects two lines and the pair of corresponding angles are equal, then the lines are parallel.
- If a transversal intersects two lines and the pair of alternate angles are equal, then the lines are parallel.
- If a transversal intersects two lines and the interior angles on the same side of the transversal are supplementary, then the lines are parallel.

**5. TRIANGLE AND ITS PROPERTIES**

**
Triangle: ** A closed figure formed by three-line segments is called a triangle.

** **In ∆ABC,

**Classification of triangles: **

Triangles can be classified according to properties their sides and angles.

**According to sides:**

Based on sides triangles are three types:

- Scanlan triangle (ii) Isosceles Triangle (iii) equilateral triangle

**According to angles:**

- Acute-angled triangle (ii) Right-angled triangle (iii) Obtuse-angled triangle

**Relationship between the sides of a triangle:**

- The sum of the lengths of any two sides of a triangle is greater than the third side.

- The difference of the lengths of any two sides of a triangle is less than the third side.

**The altitude of a triangle:**

We can draw three altitudes in a triangle. A perpendicular line drawn from a vertex to its opposite side of a triangle is called the Altitude of the triangle.

**Median of a triangle:**

In a triangle, a line drawn from vertex to mid-point of its opposite side is called the median of the triangle.

Medians of a triangle are concurrent. We can draw three medians in a triangle.

The point of concurrence of medians is called the centroid of the triangle. It is denoted by G

**Angle-sum property of a triangle:**

Sum of he angles in a triangle is 180^{0}

∠A + ∠B + ∠C = 180^{0}

**An exterior angle of a triangle:**

When one side of the triangle is produced, the angle thus formed is called an exterior triangle.

**Exterior angle property:-** The exterior angle of a triangle is equal to the sum of two interior opposite angles. ** **

x^{0}+ y^{0} = z^{0}

**6.RATIO – APPLICATIONS**

**Ratio: **Comparison of two quantities of the same kind is called ‘Ratio.

The ratio is represented by the symbol ‘:’

If the ratio of two quantities ‘a’ and ‘b’ is a : b, then we read this as ‘a is to b’

The quantities ‘a’ and ‘b’ are called terms of the ratio.

**Proportion: **if two ratios are equal, then they are said to be in proportional.

‘a’ is called as first term or antecedent and ‘b’ is called as second term or consequent.

If a : b = c : d, then a, b, c, d are in proportion and

The product of means = the product of extremes

**Unitary method:** The method in which we first find the value of one unit and then the value of the required no. of units is known as the unitary method.

**Direct proportion: **In two quantities, when one quantity increase(decreases) the other quantity also increases(decreases) then two quantities are in direct proportion.

**Percentages: **

‘percent’ means for a hundred or per every hundred. The symbol % is used to denote the percentage.

1% means 1 out of 100, 17% means 17 out of 100.

**Profit and Loss: **

Selling price = SP; Cost price = CP

If SP > CP, then we get profit

Profit = SP – CP

SP = CP + profit

If SP < CP, then we get a loss

Loss = CP – SP

SP = CP – Loss** **

**Simple interest: **

**Principle: – **The money borrowed or lent out for a certain period is called the Principle.

**Interest:** – The extra money, for keeping the principle paid by the borrower is called interest.

**Amount:** – The amount that is paid back is equal to the sum of the borrowed principal and the interest.

Amount = principle + interest

Interest (I) =

**7.DATA HANDLING**

**Data:** The information which is in the form of numbers or words and helps in taking decisions or drawing conclusions is called data.

**Observations: **The numerical entries in the data are called observations.

**Arithmetic Mean: **The average data is also called as Arithmetic mean.

The arithmetic mean always lies between the highest and lowest observations of the data.

When all the values of data set are increased or decreased by a certain number, the mean also increases or decreases by the same number.

**Mode: **The most frequently occurring observation in a data is called Mode.

If a data has two modes, then it is called bimodal data.

**Note:** If each observation in a data is repeated an equal no. of times, then the data has no mode.

**Median: **The middle most observation in a data is called the Median.

Arrange given data in ascending or descending order.

If a data has odd no. of observations, then the middle observation is median.

If a data has even no. of observations, then the median is the average of middle observations.

**Bar graph:**

Bar graphs are made up of uniform width which can be drawn horizontally or vertically with equal spacing between them.

The length of each bar tells us the frequency of the particular item.

**Ex:**

It represents two observations side by side.

**Ex:**

** ****Pie chart: **A circle can be divided into sectors to represent the given data

**Ex:**

Budget | Amount in rupees |

Food | 1200 |

Education | 800 |

Others | 2000 |

Savings | 5000 |

Total income | 9000 |

**8.CONGRUENCY OF TRIANGLES**

**Congruent figures: **Two figures are said to be congruent if they have same shape and size.

**Congruency of line segments: ** If two-line segments have same length, then they are congruent. Conversely, if two-line segments are congruent, then they have same length.

** ****Congruency of Triangles:**

Two triangles are said to be congruent if (i) their corresponding angles are equal (ii) their corresponding sides are equal.

Ex: In ∆ ABC, ∆ DEF

∠A ≅ ∠D; ∠B≅ ∠E; ∠C ≅ ∠F

AB ≅ DE; BC≅ EF; AC ≅ DF

∴∆ABC ≅ ∆DEF

**Criterion for congruency of Triangles:**

**1.Side -Side -Side congruency (SSS): –**

If three side of a triangle are equal to the corresponding three sides of another triangle, then the triangles are congruent.

∴∆ABC ≅ ∆DEF

**2.Side -Angle -Side congruency (SAS): –**

If two sides and the angle included between the two sides of a triangle are equal to the corresponding two sides and the included angle of another triangle, then the triangles are congruent.

∴ ∆ABC ≅ ∆DEF

**3.Angle – Side -Angle congruency (ASA): –**

If two angles and included side of a triangle are equal to the corresponding two angles and included side of another triangle, then the triangles are congruent.

∴ ∆ABC ≅ ∆DEF

**4.Right angle – Hypotenuse – Side congruence (RHS): –**

** **If the hypotenuse and one side of a right-angled triangle are equal to the corresponding hypotenuse and side of the other right-angled triangle, then the triangles are Equal.

∴∆ABC ≅ ∆DEF

**9.CONSTRUCTION OF TRIANGLES**

The no. of measurements required to construct a triangle = 3

A triangle can be drawn in any of the situations given below:

- Three sides of a triangle
- Two sides and the angle included between them.
- Two angles and the side included between them.
- Hypotenuse and one adjacent side of the right-angled triangle.

**Construction of a triangle when measurements of the three sides are given:**

**Ex:** construct a triangle ABC with sides AB = 4cm, BC = 7cm and AC = 5cm

**Step of constructions:**

Step -1: Draw a rough sketch of the triangle and label it with the given measurements.

Step -2: Draw a line segment of BC of length 7cm.

Step -3: with centre B, draw an arc of radius 4cm, draw another arc from C with radius 5cm such that it intersects first at A.

Step -4: join A, B and A, C. The required tringle ABC is constructed.

**Construction of a triangle when two sides and the included angle given:**

**EX: **construct a triangle ABC with sides AB = 4cm, BC = 6cm and ∠B=60^{0}

**Step of constructions:**

Step -1: Draw a rough sketch of the triangle and label it with the given measurements.

Step -2: Draw a line segment of AB of length 4cm.

Step -3: draw a ray BX making an angle 60^{0} with AB.

Step -4: draw an arc of radius 5cm from B, which cuts ray BX at C.

Step -5: join C and A, we get required ∆ABC.

**Construction of a triangle when two angles and the side between the angles given:**

**Ex: **construct a triangle PQR with sides QR = 4cm, ∠Q= 120^{0} and ∠R= 40^{0}

**Step of constructions:**

Step -1: Draw a rough sketch of a triangle and label it with the given measurements.

Step -2: Draw a line segment QR of length 4 cm.

Step -3: Draw a ray RX, making an angle 40^{0} with QR.

Step -4: Draw a ray QY, making an angle 100^{0} with QR, which is intersects ray RX.

Step -5: Mark the intersecting point of the two rays as P. Required triangle PQR is constructed.

**Construction of a triangle when two sides and the non-included angles are given**:

**Ex: **construct a triangle MAN with sides MN = 4cm, AM = 3cm and ∠A= 40^{0}

**Step of constructions:**

Step -1: Draw a rough sketch of a triangle and label it with the given measurements.

Step -2: Draw a line segment MA of length 0f 5cm.

Step -3: Draw a ray AX making an angle 40^{0} with the line segment MA.

Step -4: With M as the centre and radius 3 cm draw an arc to cut ray AX. Mark the intersecting point as N.

Step -5: join M, N, then we get required triangle MAN.

**Construction of a right-angled triangle when hypotenuse and sides are given:**

**Ex: **construct a triangle ABC, right angle at B and AB = 4cm, Ac = 5cm

**Step of constructions:**

Step -1: Draw a rough sketch of a triangle and label it with the given measurements.

Step -2: Draw a line segment BC of length 0f 4cm.

Step -3: Draw a ray BX perpendicular to BC at B

Step -4: Draw an arc from C with radius 5cm to intersect ray BX at A.

Step -5: Join A, C, then we get required triangle ABC.

**10.ALGEBRAIC EXPRESSIONS**

**Variable: **It is a dependent term. It takes different value.

Ex: m, x, a, etc.

**Constant: ** It is an independent term. It has fixed value.

**Like terms and Un like terms: **If the terms contain the same variable with the same exponents, then they are like terms otherwise, unlike terms.

Ex: 3x, –4x, x are like terms

3x, 4y, 4 are un like terms

**Coefficient: **Coefficient is a term which the multiple of other term(s)

EX: In 5x. 5 is the coefficient of x and x is the coefficient of 5

5 is numerical coefficient

x is literal coefficient

**Expression: **An expression is a single term or a combination of terms connected by the symbols ‘+’ (plus) or ‘−’ (minus).

Ex: 2x – 3. 3x, 2 +3 – 4 etc.

**Numerical Expressions: ** If every term of an expression is constant, then the expression is called numerical expression.

Ex: 2 + 3 + 5, 2 – 4 – 7, 1 + 5 – 4 etc.

**Algebraic expression: ** If an expression at least one algebraic term, then the expression is called algebraic expression.

Ex: x + y, xy, x – 3, 4x + 2 etc.

**Note: **Plus (+) and Minus (−) separate the terms

Multiplication (×) and Division (÷) do not separate the terms.

**Types of Algebraic expressions:**

**Monomial: – **If an expression has only one term, then it is called monomial.

Ex: 2x^{2}, 3y, x, y, xyz etc.

**Binomial: **If an expression has two unlike terms, then it is called binomial.

Ex: 2x+ 3y, x^{2}+ y, x +yz^{2} etc.

**Trinomial: **If an expression has three unlike terms, then it is called trinomial. Ex: 2x+ 3y + 4xy, x^{2}+ y + z, x^{2} y +yz^{2} + xy^{2} etc.

**Multinomial: **If an expression has more than three unlike terms, then it is called multinomial.

Ex: 2x+ 3y + 4xy +5, x^{2}+ y + z – 4y + 6 ,

x^{2} y +yz^{2} + xy^{2} – 4xy + 8yz etc.

**Degree of monomial: **The sum of all exponents of the variables present in a monomial is called degree of the monomial.

Ex: Degree of 5xy^{3 }

Exponent of x is 1 and exponent of y is 3** **

Sum of exponents = 1 + 3 = 4

∴ degree of 5xy^{3} is 4

**Degree of an Algebraic Expression: **The highest exponent of all the terms of an expression is called the degree of an Algebraic expression.

** Ex: **degree of x^{2} + 3x + 4x^{3 }is 3

degree of 3xy + 6x^{2}y + 5x^{2}y^{2} is 4

**Addition of like terms: **

The sum of two or more like terms is a like term with a numerical coefficient is equal to the sum of the numerical coefficients of all the like terms in addition.

Ex: 3x + 2x = (3 + 2) x = 5x

4x^{2}y + x^{2}y = (4 + 1) x^{2}y = 5x^{2}y

**Subtraction of like terms: **

The difference of two like terms is a like term with a numerical coefficient is equal to the difference between the numerical coefficients of the two like terms.

Ex: 3x − 2x = (3 − 2) x = x

4x^{2}y −2 x^{2}y = (4 −2) x^{2}y = 2x^{2}y

**Note: (**i) addition and subtraction is not done for unlike terms.** (ii) **If no terms of an expression are alike then it is said to be in the simplified form.

**Standard form of an Expression: **

In an expression, if the terms are in such away that the degree of the terms are in descending order, then the expression is said to be in standard form.

Ex: 5 – 2x^{2} + 4x +3x^{3}

Standard form is 3x^{3} – 2x^{2} + 4x + 5

**Finding the value of an expression: **

Example: find the value of expression x^{3} + y + 3, when x = 1 and y = 2

Sol: given expression is x^{3} + y + 3

Substitute x = 1 and y = 2 in above expression

(1)^{3} + 2 + 3 = 1 + 2 + 3 = 6

**Addition of algebraic expressions:**

Addition of expressions can be obtained by adding like terms.

This is in two ways: (i) Column or Vertical method (ii) Row or Horizontal method.

**Column or Vertical method: **

**Step –1: **Write the expression in standard form if necessary.

**Step –2:** write one expression below the other such that the like terms come in the same column.

**Step –3: **Add the like terms column wise and write the result just below concerned column.

**Ex: **Add x^{2} + 3x + 5, 3 – 2x + 3x^{2} and 3x – 2

**Row or Horizontal method.**

**Step –1: **Write the expression in standard form if necessary.

**Step –2:** Re-arrange them term by grouping the like terms together.

**Step –3: **Simplify the coefficients.

**Step –4: **Write the resultant expression in standard form.

** Ex: **Add x^{2} + 3x + 5, 3 – 2x + 3x^{2} and 3x –2

Sol: (x^{2} + 3x + 5) + (3 – 2x + 3x^{2}) + (3x –2)

= (x^{2 }+ 3x^{2}) + (3x – 2x + 3x) + (5 + 3 – 2)

= (1 + 3) x^{2} + (3 – 2 + 3) x + 6

=4x^{2} + 4x + 6

**Additive inverse of an expression:**

For every algebraic expression there exist another algebraic expression such that their sum is zero. These two expressions are called the additive inverse of the each other.

**Subtraction of algebraic expressions:**

This is in two ways: (i) Column or Vertical method (ii) Row or Horizontal method.

**Column or Vertical method: **

**Step –1: **Write the expression in standard form if necessary.

**Step –2:** write one expression below the other such that the expression to be subtracted comes in the second row and the like terms come one below the other.

**Step –3:** Change the sign of every term of the expression in the second row to get the additive inverse of the expression.

**Step –4: **Add the like terms column wise and write the result just below concerned column.

**Ex: **Subtract: x^{2} + 3x + 5 from 3x^{2} + 4x – 3

Sol:

**Row or Horizontal method: **

**Step –1: **Write the expressions in one row with the expression to be subtracted in a bracket with assigning negative sign to it.

**Step –2:** Add the additive inverse of the second expression to the first expression.

**Step –3: **Group the like terms and add or subtract.

**Step –4: **Write the resultant expression in standard form.

** Ex: **Subtract: x^{2} + 3x + 5 from 3x^{2} + 4x – 3

Sol: 3x^{2} + 4x – 3 – (x^{2} + 3x + 5)

= 3x^{2} + 4x – 3 – x^{2} – 3x – 5

= (3 – 1) x^{2} + (4 – 3) x + (– 3 – 5)

= 2x^{2} + x – 8

**11.EXPONENTS**

We know that,

a × a = a^{2} (a raised to the power of 2)

a × a × a = a^{3} (a raised to the power of 3)

a × a × a × a × a × a ×…. m times = a^{m}

a^{m} is in exponential form

a is called base, m is called exponent or index.

**Laws of exponents:**

- a
^{m}× a^{n}= a^{m + n} - (a
^{m})^{n}= a^{mn} - a
^{m}= a^{n}⇒ m = n - (ab)
^{m}= a^{m}.a^{n} - a
^{0 }= 1

**Standard form: ** A number which is expressed as the product of the largest integer exponent of 10 and a decimal number between 1 and 10 is said to be in standard form.

Ex: 1324 in standard form is 1.324 × 10^{3}.

**12.QUADRILATERALS**

**Quadrilateral: **A Quadrilateral is a closed figure with four sides, four angles and four vertices.

In Quadrilateral ABCD

- AB, BC, CD, and AD are sides.
- A, B, C and D are the vertices.
- ∠ABC, ∠BCD, ∠CDA and ∠DAC are the angles.

**Diagonal of a Quadrilateral:**

The line segment joining the opposite vertices of a quadrilateral are called the diagonals of the Quadrilateral. In the above figure AC, BD are the diagonals.

**Adjacent sides of a Quadrilateral:**

The two sides of a Quadrilateral which have a common vertex are called the adjacent sides of the Quadrilateral. From the above figure, AB, BC; BC, CD; CD, DA and DA, AB are the adjacent sides.

**Adjacent angles of a Quadrilateral:**

The two angles of a Quadrilateral which have a common side are called the adjacent angles of the Quadrilateral. From the above figure, ∠A, ∠B; ∠B, ∠C; ∠C, ∠D and ∠D, ∠A are the adjacent angles.

**Opposite sides of a Quadrilateral:**

The two sides of a quadrilateral, which do not have a common vertex are called opposite sides of a quadrilateral. From the above figure, AB, CD; BC, DA are the opposite sides.

**Opposite angles of a Quadrilateral:**

The two angles of a quadrilateral, which do not have a common side are called opposite angles of a quadrilateral. From the above figure, ∠A, ∠C; ∠B, ∠D are the opposite angles.

**Interior and exterior of a Quadrilateral:**

In a Quadrilateral ABCD, S, N are interior points, M, P are exterior points and A, B, C, D and Q are lies on the Quadrilateral.

**Convex Quadrilateral:**

A Quadrilateral is said to be convex Quadrilateral if all line segments joining points in the interior of the Quadrilateral also lie in the interior of the Quadrilateral.

**Concave Quadrilateral:**

A Quadrilateral is said to be concave Quadrilateral if all line segments joining points in the interior of the Quadrilateral not lie in the interior of the Quadrilateral.

**Angle sum property of a quadrilateral:**

Sum of the angle in a Quadrilateral is 360^{0}

In a Quadrilateral ABCD, ∠A + ∠B + ∠C + ∠D = 360^{0}

**Types of Quadrilaterals:**

**1.Trapezium:**

In a Quadrilateral one pair of opposite sides are parallel then it is Trapezium.

In a Trapezium ABCD, AB∥ DC; AC, BD are diagonals.

**2.Kite:**

In a Quadrilateral two distinct consecutive pairs of sides are equal in length then it is called a Kite.

In a Kite ABCD, AB = BC; AD = DC AC, BD are diagonals.

**3.Parallelogram:**

In a Quadrilateral two pairs of opposite sides are parallel then it is Parallelogram.

In a Parallelogram ABCD, AB∥ DC, AD∥ BC; AD, BD are diagonals.

Properties of parallelogram: –

- The opposite sides of a parallelogram are equal in length.
- The opposite angles are equal in measure.
- The sum of the adjacent angles is 180
^{0} - Diagonals are bisected each other and not equal in length.

**4.Rhombus:**

In a parallelogram in which two adjacent sides are equal, then it is a Rhombus.

In a Parallelogram ABCD, AB∥ DC, AD∥ BC; AD, BD are diagonals.

Properties of Rhombus: –

- All sides of a Rhombus are equal in length.
- The opposite angles are equal in measure.
- The sum of the adjacent angles is 180
^{0} - Diagonals are bisected each other perpendicularly and not equal in length.

**5.Rectangle:**

In a parallelogram all angles are equal, then it is a Rectangle.

Properties of Rectangle: –

- The opposite sides are equal in length.
- Each angle is 90
^{0}. - The sum of the adjacent angles is 180
^{0} - Diagonals are bisected each other and not equal in length.
- Each diagonal divides the rectangle into two congruent triangles.

**6.Square:**

In a rectangle adjacent sides are equal, then it is a Square.

Properties of Square: –

- All sides a Square are equal in length.
- Each angle is 90
^{0}. - The sum of the adjacent angles is 180
^{0} - Diagonals are bisected each other and equal in length.
- Each diagonal divides the Square into two congruent triangles.

## 13.AREA AND PERIMETER

**Area of a parallelogram:**

Area of parallelogram (A) = b × h square units.

The area of the parallelogram is equal to the product of its base (b) and the height(h)

**Area of a Triangle:**

Area of triangle = ½ b × h square units.

The area of the triangle is equal to half the product of its base (b) and height (h).

In a Right-angled triangle two of its sides can be the height.

Area of a Rhombus:

The area of the Rhombus is equal to half the product of its diagonals

Area of rhombus = ½ d_{1} × d_{2} square units.

**Circumference of the circle:**

Circumference of circle = 2πr = πd

**Area of rectangular path:**

Area of Rectangular path = area of outer rectangle – are of inner rectangle

**14.UNDERSTANDING 2D AND 3D SHAPES**

**Net: **Net is a short of skeleton-outline in 2d, which when folded the result in 3d shape.

**Nets of 3D shapes:**

**1.Cube:**

**2.Cylinder:**

** 3.****Pyramid:**

**Oblique Sketches: **

Oblique sketches are drawn on a grid paper to visualise 3D shapes.

**Ex:** Draw oblique sketch of a 3×3×3 cube

**Step-1: **Draw the front face

**Step-2:** Draw the opposite face, which is same as front face. The sketch is somewhat off – set from Step-1

**Step-3: **Join the corresponding corners.

**Step-4: **Redraw using dotted lines for hidden edges.

**Isometric Sketches: **

Isometric sketches are drawn on a dot isometric paper to visualise 3D shapes.

**Ex:** Draw oblique sketch of a 2×3×4 cuboid

**Step-1: **Draw a rectangle to show the front face.

**Step-2: D**raw four parallel line segments of length 3cm**.**

** ****Step-3: **Connect the corresponding corners with appropriate line segments

** ****Step-4: **This is an Isometric sketch of a cuboid

**15.SYMMETRY**

**Line of symmetry: **The line which divides a figure in to two identical parts is called the line of symmetry or axis of symmetry.

An object can have one or more than one lines of symmetry.

**Regular polygon:**

If a polygon has equal sides and equal angles, then the polygon is called Regular polygon.

**Lines of symmetry for Regular polygons:**

Regular polygon |
No. of sides |
No. of axes of symmetry |

Triangle | 3 | 3 |

Square | 4 | 4 |

Pentagon | 5 | 5 |

Polygon | n | n |

** ****Rotational symmetry: **If we rotate a figure, about a fixed point by a certain angle and the figure looks exactly the same as before, then the figure has rotational symmetry.

**Angle of rotational symmetry: ** The minimum angle of rotational of a figure to get exactly the same figure as original is called the angle of rotational symmetry or angle of rotation.

The angle of rotational of equilateral triangle is 120^{0}

The angle of rotational of a square is 90^{0}

**Order of rotational symmetry: **

The no. of times a figure, rotated through its angle of rotational symmetry before it comes to original position is called order of rotational symmetry.

The order of rotational symmetry for an equilateral triangle is 3.

The order of rotational symmetry for a square is 4.

**Note: **All figures have rotational symmetry of order 1 , as can be rotated completely through 360^{0} to come back to its original position.

An object has rotational symmetry, only when the order of symmetry is more than 1.

• Some shapes have line of symmetry and some have rotational symmetry and some have both.

Square, Equilateral triangle and Circle have both line and rotational symmetry.

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