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Class 7 Maths Integers Introduction
Natural Numbers
Numbers that come naturally to us are Natural numbers.For example, 1,2,3, 4,…
Count the number of fingers in your hand 1,2,3,4……10, these are natural numbers.
What exist below 1?
Suppose you have 5 chocolates and now you give one chocolate to one of your friend, now you are left with four, similarly you distribute remaining four
to other friends, hence you are left with no chocolate or zero chocolate.
Whole Numbers
Numbers starting from 0,1,2,3,4,5,6 ............. are called Whole numbers
Note
: The whole numbers start with 0 while natural numbers start with 1,2,3,4..
What exist below 0?
The numbers below 0 are -1, -2, -3,-4,-5,-6,-7……….. .
Examples: Suppose you borrow one chocolate from your elder brother then you will have one chocolate and that should be counted as -1.
In the Antarctica region the temperature goes well below 0, the temperature usually over there is -10
Class 7 Maths Integers What are integers
What are integers? Collection of all positive(1,2,3,4) and negative numbers(-1,-2,-3..) including 0 are theintegers. Note: 0 is neither a positive nor a negative integer.
Question: Which number is larger?
10 or 18
-10 or 18
-10 or -18
Solution:
a) 18 is larger than 10.
b) 18 is larger than -10
c) -10 is larger than -18,
These can be proved on a number line
Class 7 Maths Integers Integer on number Line
I
nteger on number Line
Note
25 > 15 (as 25 lies more on right hand side).
: The numbers on right hand side are always bigger than the numbers on the left hand side.
For example:
-25 < 15 (as 15 lies more on right hand side).
-15 > -25 (as -15 lies more on right hand side).
To represent a number on number line we mark the circle on the number line, suppose you need to represent -2, 0 and 2 on number line you put
circle over the number.
Rules for Operation on number line
To Add a +ve number, move right
To Add a –ve number, move left
To Subtract a +ve number, move left
To Subtract a –ve number, move right
Ex: 5+(-3) = 2 (From 5 move 3 jumps on left side, we get 2).
4+(-5)=-1 (From 4 move 5 jumps on left side, we get 2).
0+(-8)=-8 (From 0 move 8 jumps on left side, we get -8)
6+8= 14 (From 6 move 8 jumps on left side, we get
14) Tip: Subtract the number put the sign of bigger number.
Class 7 Maths Integers Properties of Integer
Properties of Intege
r
Closure Property
:
For any two integers a and b, a+b is always an integer
For any two integers a and b, a-b is always an integer
Hence, Addition and Subtraction follows closure property.
Example: 1+(-15)=14
1+15=16
2-(-5)=7
2-5= -3 .
Commutative Property
i.e. a-b b-a.
:For any two integers a and b, a+b=b+a
But, this is NOT true for Subtraction.
Example:
Statement 1 Statement 2 Inference
3 + 4 =7 4 + 3 = 7 Both statements are equal
–15 + (–10) = -25 (–10) + (–15) = -25 Both statements are equal
3 + 12 =15 12 + 3 = 15 Both statements are equal
3-(-5)=9 -5-3=-8 Both statements are different
2-5=-3 5 – 2=3 Both statements are different
Associative Property
:
For three integers a, b and c, [a+b]+c = a+[b+c]
But, this law doesn’t hold true for Subtraction.
i.e. a-(b-c) (a-b)-c
Example:
Statement 1 Statement 2 Result
2+[5+3]=10 [2+5]+3=10 Both statementsare equal
8+[-2+(-3)]=3 [8+(-2)]+(-3)=3 Both statementsare equal
8-[(-2)-(-3)] = 8-(-2+3) = 7 [8-(-2)]-(-3) = 10-(-3) = 13 Both statementsare different
Class 7 Maths Integers Multiplication of an integer Multiplication of an integer
Multiplication of positive and negative intege
We should remember when we multiply,
r
a xb = ab i.e two positive integer when multiply gives positive integer.
(-a)x(-b)=ab i.e two negative integer when multiply gives positive number.
(-a)x(b)=-ab i.e one positive and one negative integer when multiply gives negative number.
Tip: Find the product then a give sign according to the case mentioned above.
Question: Multiply the following numbers
(-2)x(-3)
(-3)x6
2x4
4x(-6)
Solution: a. (-2)x(-3) = 6
b. (-3)x6 = -18
c. 2x4 = 8
d. 4x(-6) = -24
Tip: Multiply the numbers then put the sign accordingly
Negative-positive-negative gives positive result
Example: -2x3x(-4)=24
Multiplication of three or more integers
Negative-Negative-Negative gives Negative result
Example: -2x(-3)x(-4)= - 24
Negative-positive-negative gives positive result
Example: -2x3x(-4)=24
Note: If the numbers of negative sign is even then the sign is positive and if the number of negative sign is odd then the sign of a number is negative.
Multiplication by
0:
Any number when multiplied with 0 always gives 0 value.
Example: 0x6=0
0x5=0
0x(-2)=0
Multiplicative Identity(1): When we multiply 1 with the number the result will be the number itself i.e. ax1=a.
Example: 1 x 3 = 3
2 x 1 = 2
3 x 1 = 3
-10 x 1 = -10
Note: When we multiply -1 with the number the sign changes.
Example: 2x -1 = -2
-2 x -1 = 2
Class 7 Maths Integers Multiplication Property Multiplication Property
:
Closure under Multiplication: If we multiply two integers “a, b” the product of two integers will always an integer.
The below example proves that.
Statement-1 Inference
2x3=6 Result is Integer
3x(-7)=-21 Result is Integer
-2x(-10)=20 Result is Integer
Commutative of Multiplication
For any two integers a and b, axb= bxa.
Example:
:
Statement 1 Statement 2 Inference
3× (– 4) = –12 (– 4) × 3 = –12 Both statementsare equal
(–15) × (–10) = 150 (–10) × (–15) = 150 Both statementsare equal
(–30) × 12 = -360 12 × (–30) = -360 Both statementsare equal
Associative property of Multiplication
If you are multiplying three integers a, b and c, (a x b) x c = a x (b x c)
Example:
:
Statement 1
Statement 2
Inference
[(–3) × (–2)] ×5 = 30
(–3) × [(–2) × 5] = 30
Both statements are equal
[(7) × (– 6)] × 4 = -168
7 × [(– 6) × 4] = -168
Both statements are equal
Distributive Property:.
For three integers a, b and c,
a x (b + c) = a x b + a x c
a x (b - c) = a x b – a x c
Example:
[(7) × (3)] × 2 = 42 7 × [(3) × 2] = 42 Both statements are equal
Statement 1 Statement 2 Inference
18 × [7 + (–3)] = 72 [18 × 7] + [18 × (–3)] = 72 Both statements are equal
(–21) × [(– 4) + (– 6)] = 210 [(–21) × (– 4)] + [(–21) × (–
6)]=210
Both statements are equal
18 × [7 -6] = 18 [18 × 7] - [18 × (6)] = 18 Both statements are equal
Class 7 Maths Integers Division of an Integer
Division of an Integer
:
Division refers to splitting into equal parts.
Example Dividing 4 ice-creams into two parts,
We get,
2 ice-creams in each part
i.e 4/2=2
Division of a negative integer by a positive integeWhen we divide a positive integer by a negative integer, we first divide them as whole numbers and then put a minus sign (–) before the quotient. That
is, we get a negative integer.
r
In general, for any two positive integers a and b
a (-b)= (-a) b where b 0
Example: 125 (-25) = -5
-64 8 = 8
Division by zero(0
)
When we divide a number by zero the result is undefined i.e.a 0= undefined
When 0 is divided by any number the result is zero(0) itself
Example: 5 0 = undefined
0 7= 0
0 (-2)= 0
Division by one(1)
When we divide a number by 1 the result is the number itself
Example: 27 1 =27
-12 1 = -12
20 1 =20
Class 7 Maths Integers Division Property
Division Property:
Commutative Property: In Division“a b b a”. Hence, division does not follow Commutative Property
Example: -15 3 = -5 But, 3 = 1/5 they are not equal
Associative Property: Division does not follow Associative Property.
Example: 15 [(-3) (-3)] = 15 1 = 5 But, [15 (-3)] (-3) = 5/3Both statements are not equal
Distributive Property: For three integers “a [b+ c] a b + c”.
Hence division does not follow Distributive Property.
Example: 15 [3+2] = 15 5 = 3 But, 15 +15 2 =5+ 15/2
Both statements are not equal
Fractions and Decimals
Multiplication and Division on Fraction
Divisions of Fractions
To obtain the reciprocal of a fraction, interchange the numerator with denominator. To divide a whole number by a fraction, take the reciprocal of the fraction and then multiply it with the whole number. To divide a fraction by a whole number, multiply the fraction with the reciprocal of the whole number. To divide a fraction by a fraction, multiply the first fraction with the reciprocal of the second fraction.
Multiplication and Division on Decimals
Multiplication of Decimals
To multiply a whole number by a decimal number, follow these steps: Ignore the decimal and multiply the two numbers. Count the number of digits to the right of decimal point in the original decimal number. Insert the decimal, from right to left, in the answer by the same count. To multiply a decimal number by a decimal number, follow these steps: Ignore the decimals and multiply the two numbers. Count the number of digits to the right of decimal point in both the decimal numbers. Add up the number of digits counted and insert the decimal, from right to left, in the answer by the same count.
To multiply a decimal number with 10, 100 or 1000, follow these steps:
While multiplying a decimal number with 10, retain the original number and shift the decimal to the right by one place.
While multiplying a decimal number with 100, retain the original number and shift the decimal to the right by two places. While multiplying a decimal number with 1000, retain the original number and shift the decimal to the right by three places.
Division of Decimals
To divide a decimal number by a whole number, follow these steps: Convert the decimal number into a fraction. Take the reciprocal of the divisor. Multiply the reciprocal with the fraction. To divide a decimal number by another decimal number, follow these steps: Convert both the decimal numbers into fractions. Take the reciprocal of the divisor. Multiply the reciprocal with the fraction.
Data Representation
A bar graph is a visual representation or organized data. A bar graph consists of bars which have uniform width. The lengths of the bars depend on the frequency or the scale you choose. The double bar graph helps in comparing two data sets. The likelihood of getting an outcome is known as probability.
Step – 2: Tabulate the data in a frequency distribution table.
Step – 3: The most frequently occurring observation will be the mode.
Median
Median refers to the value that lies in the middle of the data with half of the observations above it and the other half of the observations below it. The following are the steps to calculate median.
Data Handling (Mean, Median And Mode)
Mean
Arithmetic mean is a number that lies between the highest and the lowest value of data. Note that we need not arrange the data in ascending or descending order to calculate arithmetic mean. Range = Highest observation – Lowest observation
Mode
Mode refers to the observation that occurs most often in a given data.
The following are the steps to calculate mode:
Step – 1: Arrange the data in ascending order.
x
Step – 2:
Step – 1: Arrange the data in ascending order.
The value that lies in the middle such that half of the observations lie above it and the other half below it will be the median.
T he mean, mode and median are representative values of a group of observations or data, and lie between the minimum and maximum values of the data. They are also called measures of the central tendency.
Simple Equations : Mathematics
Introduction to Simple Equations An equation is a condition of equality between two mathematical expressions. e.g. 2x - 3 = 5, 3x + 9 = 11, 4y + 2 = 12
The value of the variable for which the left hand side of an equation is equal to its right hand side is called the solution of that equation. e.g. For the equation, 5x + 5 = 15, x = 2 is a solution.
When the same number is added or subtracted to or from both the sides of an equation, the value of the left hand side remains equal to its value on the right hand side. e.g. (1) 5x + 3 = 13
On adding 2 to both sides of the equation, we get 5x + 3 + 2 = 13 + 2 5x + 5 = 15
(2) On subtracting 2 from both sides of the equation, we get 5x + 3 - 2 = 13 - 2 5x + 1 = 11
When an equation is divided or multiplied on both the sides by a non-zero number, the value of the left hand side remains equal to its value on the right hand side. e.g. (1) 5x + 3 = 13
On dividing both sides of the equation by 4, we get (5x + 3) ÷ 4 = 13 ÷ 4
2) 5x + 1 = 13 On multiplying both sides of the equation by 4, we get 4(5x + 1) = 4(13) 20x + 4 = 52
Application Of Simple Equations
To find the solution of an equation, a series of identical mathematical operations are performed on both the sides of the equation so that only the variable remains on one side. On simplifying all the numbers, the result obtained is the solution of the equation. Ex: 3x + 8 = 83 3x + 8 - 8 = 83 - 8 3x = 75 x = 75 3 x = 25.
Moving a term of an equation from one side to the other side is called transposing. Transposing a number is same as adding to or subtracting the same number from both sides of the equation. Ex: Solve 2x + 8 = 24 Given, 2x + 8 = 24 Transposing 8 to the right hand side, we get ⇒ 2x = 24 - 8 ⇒ 2x = 16 ⇒ x = 16 2 ⇒ x = 8. Hence, the value of x is 8.
The sign of a number changes when it is transposed from one side of the equation to the other.
Lines and Angles : Mathematics
Angles
An angle is formed when two lines or line segments meet. Complementary angles are a pair of angles, the sum of whose measure is equal to 90°. Supplementary angles are a pair of angles, the sum of whose measure is equal to 180°. Adjacent angles have a common vertex, a common arm and non-common arms are on either side of the common arm. Adjacent angles have no common interior points. A linear pair is a pair of adjacent angles whose non-common sides are opposite rays. The angles in a linear pair are supplementary. Vertically opposite angles are opposite to each other, and these angles are equal in measure.
Pairs of Lines
Lines that meet at a point are called intersecting lines. Lines that always remain the same distance apart and never meet are called parallel lines. A line that intersects two or more lines at a distinct point is called a transversal. When two lines re intersected by a transversal, pairs of corresponding angles, alternate angles and interior angles on the same side of the transversal are formed. Angles formed on the same side of the transversal, on the same side of the two lines and at corresponding vertices are called corresponding angles. When two lines are intersected by a transversal, the pairs of angles on opposite sides of the transversal at the two distinct points of intersection and between the two lines are called alternate interior angles.
When two lines are intersected by a transversal, the pairs of angles on opposite sides of the transversal at the two distinct points of intersection but outside the two lines are called alternate exterior angles. Angles that have different vertices lie on the same side of the transversal and are interior angles are called consecutive interior angles or allied or co-interior angles. If two parallel lines are cut by a transversal then each pair of interior angles on the same side of transversal are supplementary, each pair of corresponding angles are equal and each pair of alternate interior angles are equal. When a transversal cuts two lines such that pairs of corresponding angles are equal, the lines are parallel. When a transversal cuts two lines such that pairs of alternate interior angles are equal, the lines are parallel. When a transversal cuts two lines such that pairs of interior angles on the same side of the transversal are supplementary, the lines are parallel.
The Triangle and Its Properties
A triangle is a closed figure made of three line segments. Every triangle has three sides, three
angles, and three vertices. These are known as the parts of a triangle. The sides and the angles of every triangle differ from one another; therefore, they do not look alike. Triangles can be classified based on their sides and angles.
Based on their sides, there are equilateral,isosceles and scalene triangles. Based on their angles, there are acute, obtuse and right-angled triangles.
Equilateral triangle: A triangle in which all the sides are equal is called an equilateral triangle. All the three angles of an equilateral triangle are also equal, and each measures 60°.
Isosceles triangle: A triangle in which any two sides are equal is called an isosceles triangle. In an isosceles triangle, the angles opposite the equal sides are called the base angles, and they are equal.
Scalene triangle: A triangle in which no two sides are equal is called an Scalene triangle.
Acute-angled triangle: A triangle with all its angles less than 90° is known as an acute-angled triangle.
Obtuse-angled triangle: A triangle with one of its angles more than 90° and less than 180° is known as an obtuse-angled triangle.
Right-angled triangle: A triangle with one of its angles equal to 90° is known as a right-angled triangle. The side opposite the 90° angle is called the hypotenuse, and is the longest side of the triangle.
Mark the mid-point of the side of a triangle, and join it to its opposite vertex. This line segment is called a median. It is defined as a line segment drawn from a vertex to the mid-point of the opposite side. You can draw three medians to a given triangle. The medians pass through a common point. Hence, the medians of a triangle are concurrent. This point of concurrence is called the centroid, and is denoted b y G. The centroid and medians of a triangle always lie inside the triangle. The centroid of a triangle divides the median in the ratio 2:1.
Altitude: The altitude of a triangle is a line segment drawn from a vertex and is perpendicular to the opposite side. A triangle has three altitudes. The altitudes of a triangle are concurrent. The point of concurrence is called the orthocentre, and is denoted by O. The altitude and orthocentre of a triangle need not lie inside the triangle.
Properties of Triangles
An exterior angle of a triangle is equal to the sum of its interior opposite angles. The total measure of the three angles of a triangle is 180°. Sum of the length of any two sides of a triangle is greater than the length of the third side. In a right-angled triangle, the side opposite to the right angle is called the hypotenuse and the other two sides are called its legs. The Pythagoras Property states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares on the legs. If the Pythagoras Property holds, the triangle must be right-angled.
Consider triangles ABC and XYZ . Cut triangle ABC and place it over XYZ. The two triangles cover each other exactly, and they are of the same shape and size . Also notice that A falls on X, B on Y, and C on Z. Also, side AB falls along XY, side BC along YZ, and side AC along XZ. So, we can say that triangle ABC is congruent to triangle XYZ. Symbolically, it is represented as
Congruence of Triangles
Congruence of Plane Figures
If two objects are of exactly the same shape and size, they are said to be congruent and the relation between the two objects being congruent is called congruence. The method of superposition examines the congruence of plane figures, line segments and angles. A plane figure is any shape that can be drawn in two dimensions. Two plane figures are congruent, if each, when superimposed on the other, covers it exactly. If two line segments have the same or equal length, they are congruent. Also, if two line segments are congruent, they have the same length. If two angles have the same measure, they are congruent. Also, if two angles are congruent, their measures are the same. If two angles are congruent, the length of their arms do not matter.
Criteria for Congruence of Triangles
Congruence of triangles:
ASA congruence criterion: Two triangles are congruent if two angles and the included side of one triangle are equal to the corresponding two angles and the included side of the other triangle.
RHS congruence criterion: Two right-angled triangles are congruent if the hypotenuse and a side of one triangle are equal to the hypotenuse and the corresponding side of the other triangle.
SAS congruence criterion: Two triangles are congruent if two sides and the included angle of one triangle are equal to the corresponding two sides and the included angle of the other triangle.
SSS congruence criterion: Two triangles are congruent if three sides of one triangle are equal to the threecorresponding sides of the other triangle.
We can tell if two triangles are congruent using 4 axioms: SAS axiom, ASA axiom, SSS axiom and RHS axiom.
In two congruent triangles ABC and XYZ, thecorresponding vertices are A and X, B and Y, and C and Z, that is, A corresponds to X, B to Y, and C to Z. Similarly, the corresponding sides are AB and XY, BC and YZ, and AC and XZ. Also, angle A corresponds to X, B to Y, and C to Z. So, we write ABC corresponds to XYZ.
So, in general, we can say that two triangles are congruent if all the sides and all the angles of one triangle are equal to the corresponding sides and angles of the other triangle.
ASA congruence criterion: Two triangles are congruent if two angles and the included side of one triangle are equal to the corresponding two angles and the included side of the other triangle.
RHS congruence criterion: Two right-angled triangles are congruent if the hypotenuse and a side of one triangle are equal to the hypotenuse and the corresponding side of the other triangle.
SAS congruence criterion: Two triangles are congruent if two sides and the included angle of one triangle are equal to the corresponding two sides and the included angle of the other triangle.
SSS congruence criterion: Two triangles are congruent if three sides of one triangle are equal to the threecorresponding sides of the other triangle.
We can tell if two triangles are congruent using 4 axioms: SAS axiom, ASA axiom, SSS axiom and RHS axiom.
The ratio of two quantities in the same unit is a fraction that shows how many times one quantity is greater or smaller than the other. When two ratios are equivalent, the four quantities are said to be in proportion.
Ratio and proportion problems can be solved by using two methods, the unitary method and equating the ratios to make proportions, and then solving the equation.
Percentages: Percentage is another method used to compare quantities. Percentages are numerators of fractions with the denominator 100.
Meaning of percentage: Per cent is derived from the Latin word ‘per centum’, which means per hundred. Per cent is represented by the symbol - %.
Ratios and Proportions
Comparing Quantities
Ratios are used to compare quantities. Ratios help us to compare quantities and determine the relation between them. We write ratios in the form of fractions, and then compare them by converting them into like fractions. If these like fractions are equal, then we say that the given ratios are equivalent.
Eg:6 pens cost Rs 90. What would be the cost 10 such pens?
Solution:
Cost of 6 pens :Rs 90 Cost of 1 pen = 90/6=Rs 15 Hence, cost of 10 pens =10×15=150
The ratio of two quantities in the same units is a fraction that shows how many times one quantity is greater/smaller than the other. To calculate the ratio of two quantities, the units must be the same.
If you are given any three of these quantities the fourth one can be calculate using the interest formula.
Application of Percentages
Step 3: Put a percent sign next to the number
Step 2: Multiply the fraction by hundred or shift the decimal point two places to the right
Step 1: Convert the decimal into a fraction
To convert a decimal into percentage:
Step 2: Divide the number by hundred of move the decimal point two places to the left in the numerator.
Step 1: Remove the percent sign.
To convert a percentage into a decimal:
Step 1: Drop the percentage sign, and then divide the number by hundred.
To convert a percentage into fraction:
Conversions
Compare different ratios to determine whether they are equivalent ratios or not. If two fractions are equal then the given ratios are equivalent. When two ratios are equivalent then the four quantities are said to be in proportion. Ratio and proportion problems can be solved by using two methods, unitary method and equating ratios to make proportions and solve the equation. Percentage means per hundred. It is another method used to compare quantities. Fractions can be compared by converting them into percentages.
Introduction to Rational Numbers
All numbers, including whole numbers, integers, fractions and decimal numbers , can be written in the numerator-denominator form. A rational number is a number that can be written in the form p/q,
where p and q are integers and q ≠ 0. .
The denominator of a rational number can never be zero. A rational number is positive if its numerator
and denominator are both either positive integers or negative integers. . If either the numerator or the denominator of a rational number is a negative integer, then the rational number is called a negative rational number.
The rational number zero is neither negative nor positive.
On the number line:
Positive rational numbers are represented to the right of 0. Negative rational numbers are represented to the left of 0.
By multiplying or dividing both the numerator and the denominator of a rational number by the same non- zero integer, we can get another rational number that is equivalent to the given rational number.
A rational number is said to be in its standard form if its numerator and denominator have no common factor other than 1, and its denominator is a positive integer.
To reduce a rational number to its standard form, divide its numerator and denominator by their Highest Common Factor (HCF). To find the standard form of a rational number with a negative integer as the denominator, divide its numerator and denominator by their HCF with a minus sign.
Construction of Triangles : Mathematics
Any one of the following sets of measurements are required to construct a triangle-
Length of the three sides Two sides and the included angle
Two angles and the included side Length of the hypotenuse and one side in case of a right-angled triangle.
Construction of a triangle when measurements of its three sides are given Construct ΔABC, when AB = 6 cm, BC = 7 cm and CA = 9 cm.
Steps of construction: Step 1: Draw line segment BC = 7 cm. Step 2: Draw an arc with B as the centre and the radius equal to 6 cm. Step 3: Draw an arc with C as the centre and the radius equal to 9 cm. Step 4: Name the point of intersection of these two arcs as A. Step 5: Join points A and B, and points A and C. Triangle ABC is the required triangle.
Construction of a triangle when measurements of two sides and the included angle are given Construct ΔPQR, when PQ = 4 cm, QR = 6 cm and ∠PQR = 60°.
Steps of construction: Step 1: Draw line segment QR = 6 cm. Step 2: Construct an angle of 60° at point Q. Step 3: Draw an arc on the ray QX with Q as the centre and the radius equal to 4 cm. Step 4: Name the point where the arc cuts ray QX, as P. Step 5: Join points P and R. Triangle PQR is the required triangle.
Construction of a triangle, when two angles and the included side are given Construct ΔXYZ, when ∠ZXY = 40°, ∠XYZ = 95° and the included side XY = 8 cm.
Steps of construction: Step 1: Draw line segment XY = 8 cm. Step 2: Construct an angle of 40° at X with XY. Step 3: Construct another angle of 95° at Y with YX. Step 4: Name the point of intersection of the two rays as Z. Triangle XYZ is the required triangle.
Construction of a right-angled triangle, when the length of one side and the hypotenuse are given Construct a right-angled triangle LMN, with hypotenuse LN = 8 cm and side MN = 5 cm.
Steps of construction: Step 1: Draw line 'l'. Step 2: Mark a point on 'l' and name it M. Step 3: Draw a line segment MN = 5 cm on 'l' . Step 4: Construct a right angle LMN at M. Step 5: Draw an arc with N as the centre and radius equal to 8 cm, such that it intersects MX. Step 6: Mark the point of intersection as L. Step 7: Join points L and N. Triangle LMN is the required triangle.
Construction of Parallel Lines : Mathematics
Two lines in a plane that never meet each other at any point are said to be parallel to each other.
Any line intersecting a pair of parallel lines is called a transversal.
Properties of angles formed by parallel lines and transversal:
All pairs of alternate interior angles are equal.
All pairs of corresponding angles are equal.
All pairs of alternate exterior angles are equal.
The interior angles formed on the same side of the transversal are supplementary (the sum of their measures is 180°).
Steps to construct parallel lines using the alternate interior angle property:
Step 1: Draw line 'l' and point A outside it. Step 2: Mark point B on line 'l'. Step 3: Draw line 'n' joining point A and point B. Step 4: Draw an arc with B as the centre, such that it intersects line 'l' at D and line 'n' at E. Step 5: Draw another arc with the same radius and A as the centre, such that it intersects line 'n' at F. Ensure that arc drawn from A cuts the line 'n' between A and B. Step 6: Draw another arc with F as the centre and distance DE as the radius. Step 7: Mark the point of intersection of this arc and the previous arc as G. Step 8: Draw line 'm' passing through points A and G.
Line 'm' is the required parallel line. Verification of the construction
If the pair of alternate interior angles are equal in measure, then line 'l' is parallel to line 'm.'
Steps to construct parallel lines using the corresponding angle property
Step 1: Draw line 'l' and point P outside it. Step 2: Mark point Q on line 'l'. Step 3: Draw line 'n' joining point P and point Q. Step 4: Draw an arc with Q as the centre, such that it intersects line 'l' at R and line 'n' at S. Step 5: Draw another arc with the same radius and P as the centre, such that it intersects line 'n' at X. Ensure that arc drawn from P cuts the line 'n' outside QP. Step 6: Draw another arc with X as the centre and distance RS as the radius, such that it intersects the previous at Y. Step 7: Draw line 'm' passing through points P and Y.
Line 'm' is the required parallel line.
Verification of the construction
If the pair of corresponding angles are equal in measure, then line 'l' is parallel to line 'm'.
A triangle is a polygon with three vertices, and three sides or edges that are line segments. A triangle with vertices A, B, and C is denoted as ABC
The perimeter of a triangle is the sum of the lengths of its sides. If the three sides are a, b, and c, then perimeter
The area of a triangle is the space enclosed by its three sides. It is given by the formula, the base and h is the altitude.
where b is
Quadrilateral A simple closed figure bounded by four line segments is called a quadrilateral.
Rectangle Square Parallelogram Rhombus
Perimeter and Area
Triangle:
Various types of quadrilateral are:
The area of a parallelogram is the product of its base and perpendicular height or altitude.
In the figure, the perimeter of parallelogram ABCD = 2(AB + BC)
The perimeter of a parallelogram is twice the sum of the lengths of the adjacent sides.
Parallelogram A quadrilateral in which both the pairs of opposite sides are parallel is called a parallelogram.
In the figure, the perimeter of square ABCD = AB2 or BC2 or CD2 or DA2.
The area of a square with side s is s2
In the figure, the perimeter of square ABCD = 4AB or 4BC or 4CD or 4DA.
The perimeter of a square with side s units is 4s.
Square A square is a quadrilateral with four equal sides, and each angle of measure 90o.
In the figure, the area of rectangle ABCD = AB x BC.
The area of a rectangle is the product of its length and breadth.
In the figure, the perimeter of rectangle ABCD = 2(AB + BC).
The perimeter of a rectangle is twice the sum of the lengths of its adjacent sides.
Rectangle A rectangle is a quadrilateral with opposite sides equal, and each angle of measure 90o.
In the figure, the area of parallelogram ABCD = AB x DE or AD x BF.
Rhombus
A parallelogram in which the adjacent sides are equal is called a rhombus.
The perimeter and area of a rhombus can be calculated using the same formula as that for a parallelogram.
Circle:
A circle is defined as a collection of points on a plane that are at an equal distance from a fixed point on the plane. The fixed point is called the centre of the circle.
Circumference: The distance around a circular region is known as its circumference.
Diameter: Any straight line segment that passes through the centre of a circle and whose end points are on the circle is called its diameter.
Radius: Any line segment from the centre of the circle to its circumference.
Circumference of a circle = , where r is the radius of the circle or , where d is the
diameter of the circle.
Circumference = Diameter x 3.14
Diameter(d) is equal to twice radius(r).
Circles with the same centre but different radius are called concentric circles.
Circle: The area of a circle is the region enclosed in the circle.
Any side of a parallelogram can be taken as the base. The perpendicular dropped on that side from the opposite vertex is known as the height (altitude).
The area of a circle can be calculated by using the formula:
if radius r is given if diameter D is given
if circumference C is given
Application of Algebraic Expressions
Comparison of Rational Numbers
While comparing positive rational numbers with the same denominator, the number with the greatest numerator is the largest. It is easy to compare these numbers if their denominators are the same. Eg: A positive rational number is always greater than a negative rational number. While comparing negative rational numbers with the same denominator, compare their numerators ignoring the minus sign. The number with the greatest numerator is the smallest.
Positive rational numbers lie to the right of 0, while negative rational numbers lie to the left of 0 on the number line.
To compare rational numbers with different denominators, convert them into equivalent rational numbers with the same denominator, which is equal to the LCM of their denominators. You can find infinite rational numbers between any two given rational numbers.
Application of Algebraic Expressions
Operations on Rational Numbers
The denominator of the sum or difference of two rational numbers with the same denominator is the same as the common denominator of the given numbers. The numerator of the sum of two rational numbers with the same denominator is the sum of the numerators of the given numbers with their correct sign. The numerator of the difference of two rational numbers with the same denominator is the difference between the numerators of the given numbers with their correct sign. To add or subtract rational numbers with different denominators, we convert them into equivalent rational numbers having common denominator equal to the LCM of the denominators of the given numbers.
Two rational numbers whose sum is zero are called additive inverse of each other. The numerator and denominator of the product of two rational numbers are equal to the product of their individual numerators and denominators. Two rational numbers whose product is 1 are called reciprocals of each other. A rational number and its reciprocal always have the same sign. To divide one rational number by a second rational number, we actually multiply the first number by the reciprocal of the second number.
Exponent and Power : Mathematics
A n exponent or power is a mathematical representation that indicates the number of times that a number is multiplied by itself. If a number is multiplied by itself m times, then it can be written as: a x a x a x a x a...m times = am
Here, a is called the base, and m is called the exponent, power or index. Numbers raised to the power of two are called square numbers.
Square numbers are also read as two-square, three-square, four-square, five-square, and so on. Numbers raised to the power of three are called cube numbers. Cube numbers are also read as two-cube, three-cube, four-cube, five-cube, and so on. Negative numbers can also be written using exponents. If an = b, where a and b are integers and n is a natural number, then an is called the exponential form of b. The factors of a product can be expressed as the powers of the prime factors of 100. This form of expressing numbers using exponents is called the prime factor product form of exponents. Even if we interchange the order of the factors, the value remains the same. So a raised to the power of x multiplied by b raised to the power of y, is the same as b raised to the power of y multiplied by a raised to the power of x. The value of an exponential number with a negative base raised to the power of an even number is positive. If the base of two exponential numbers is the same, then the number with the greater exponent is greater than the number with the smaller exponent. A 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 known as its standard form.
Laws of Exponents Multiplication of Powers with the Same Base
When numbers with the same base are multiplied, the power of the product is equal to the sum of the powers of the numbers.
If 'a' is a non-zero integer, and 'm' and 'n' are whole numbers then, am × an = am+n.
Division of Powers with the Same Base
When numbers with the same base are divided, then the power of the quotient is equal to the difference between the powers of the dividend and the divisor.
If 'a' is a non-zero integer, and 'm' and 'n' are whole numbers then, am ÷ an = am-n.
Power of a Power
If 'a' is any non-zero integer, and ‘m’and ‘n’ are whole numbers then, (am)n = amn.
Multiplication of Powers with the Same Exponent
If 'a' is any non-zero integer, and ‘m’ is a whole number then, am × bm = (ab)m.
Division of Powers with the Same Exponent
If a and b are any non-zero integers and m is a whole number then, am ÷ bm = ( a b )m.
Numbers with an exponent of zero
For any non-zero integer a, a0 = 1.
Line Symmetry
The word symmetry comes from the Greek word symmetros, which means even. A figure has line symmetry if a line can be drawn dividing it into two identical parts. The line is called the line of symmetry or axis of symmetry.
For a line segment, the perpendicular bisector is the line of symmetry. For an equilateral triangle, the bisectors of the internal angles are the lines of symmetry. For a square, the lines of symmetry are the diagonals and the lines joining the mid-points of the opposite sides. The lines of symmetry of a rectangle are the lines joining the mid-points of the opposite sides. The line of symmetry of an isosceles triangle is the perpendicular bisector of the non-equal side. A scalene triangle, in which all the sides are of different lengths, doesn’t have any line of symmetry.
Line symmetry is also known as reflection symmetry because a mirror line resembles the line of symmetry, where one half is the mirror image of the other half. Remember, while looking at a mirror, an object placed on the right appears to be on the left, and vice versa.
Most irregular polygons do not have line symmetry. However, some of them do. Look at the rectangle and the isosceles triangle. A rectangle has two lines of symmetry, and an isosceles triangle has one line of symmetry. Some letters have line symmetry. The letters A, B, C, D, E, I, K, M, T, U, V, W and Y have one line of symmetry. The letter H overlaps perfectly both vertically and horizontally. So it has two lines of symmetry. Similarly, the letter X has two lines of symmetry. The letters F, G, J, L, N, P, Q, R, S and Z have no line of symmetry.
Irregular polygon:
A polygon is said to be a regular polygon if all its sides are equal in length and all its angles are equal in measure. If a polygon is not a regular polygon, then it is said to be an irregular polygon. Regular and irregular polygons have lines of symmetry. An equilateral triangle is regular because each of its sides has the same length, and each of its angles measures sixty degrees. The number of lines of symmetry in a regular polygon is equal to the number of sides that it has. A pentagon has five lines of symmetry. Similarly, a regular octagon has eight sides, and therefore, it will have eight lines of symmetry, while a regular decagonhas ten sides, so it will have ten lines of symmetry.
Regular polygon:
About Rotational Symmetry
Any object or shape is said to have rotational symmetry if it looks exactly the same at least once during a complete rotation through three hundred and sixty degrees. During the rotation, the object rotates around a fixed point. Its shape and size do not change. This fixed point is called the centre of rotation. Rotation may be clockwise or anti-clockwise. A full turn refers to a rotation of three hundred and sixty degrees. A half turn refers to a rotation of one hundred and eighty degrees. A quarter turn refers to a rotation of ninety degrees. The angle at which a shape or an object looks exactly the same during rotation is called the angle of rotation. The order of rotational symmetry can be defined as the number of times that a shape appears exactly the same during a full 360o rotation.
The centre of rotation of a square is its centre. The angle of rotation of a square is 90 degrees, and its order of rotational symmetry is 4.
Symmetry can be seen in the English alphabet as well. The letter H has both line symmetry and rotational symmetry.
There are many shapes that have only line symmetry and no rotational symmetry at all. Some objects and shapes have both, line symmetry as well as rotational symmetry. The Ashok Chakra in the Indian national flag has both, line symmetry and rotational symmetry.
The centre of rotation of a circle is the centre of the circle .
Visualising Solid Shapes
Introduction to Solid Shapes
Plane or to dimensional figures have only length and breadth and they lie in a single plane whereas three dimensional solids have length, breadth and height and they do not lie entirely on a plane. The flat surfaces that form the skin of solid are called its faces, the line segments that form the skeleton are called edges and the points where the edges meet are called vertices. All two dimensional figures can be identified as the faces of three dimensional solid shapes. The net of a three dimensional solid is a two dimensional skeleton outline, which when folded results in the three dimensional shape. Solid shapes can be drawn on a flat surface, which is known as the two dimensional representation of a three dimensional solid. Sketches of solid are two types; oblique and isometric. Oblique sketches are drawn on squared paper. They do not have exact lengths but still covey all the significant aspects of the appearance of a solid. Isometric sketches are drawn on dotted or isometric sheets and have the exact measurements of solids. Viewing the different section of Solids Three dimensional objects are solids have lengths, breadth and height and look different from various locations. Sections of a solid can be viewed in a number of ways. Visualizing a solid help to analyse or see the hidden parts of the solid. A solid can be viewed from different angles. Viewing a solid from the front, side and top are three most common ways of viewing the solid. Cutting or slicing a solid with result in its cross-section, which is also one way of viewing the section of a solid. Observing the two dimensional shadow of a three dimension solid is also a method of viewing a solid. Shadows of solids are of different sizes depending on the position of the solid and he position of the source of light.
