Category: CBSE

Class 7 Maths Notes students can refer to the Exponents and Powers Class 7 Notes Maths Chapter 13 https://www.cbselabs.com/exponents-and-powers-class-7-notes/ Pdf here. They can also access the CBSE Class 7 Exponents and Powers Chapter 13 Notes while gearing up for their Board exams.

CBSE Class 7 Maths Notes Chapter 13 Exponents and Powers

Exponents And Powers Class 7 Notes

Exponents
We can write large numbers in a short form using exponents.
For example: 10,000 = 10 × 10 × 10 × 10 = 10⁴
Here, ‘10’ is called the base and ‘4’ the exponent. The number 10⁴ is read as 10 raised to the power of 4 or simply as the fourth power of 10.
10⁴ is called the exponential form of 10,000.

(1)^{any natural number} = 1

(-1)^{an odd natural number} = -1

(-1)^{an even natural number} = +1

a^m × aⁿ = a^m+n, where m and n are whole numbers and a (≠ 0) is an integer.
This formula can be used to write answers to above questions.

For any non-zero integer a,
a^m ÷ aⁿ = a^m-n where m and n are whole numbers and m > n.

For any non-zero integer a,
(a^m)ⁿ = a^mn (where m and n are whole numbers)

For any non-zero integer a
a^m × b^m = (ab)^m (where m is any whole number)

(where m is a whole number; a and b are any non-zero integers)

a⁰ = 1 (for any non-zero integer a)
Any number (except 0) raised to the power (or exponent) 0 is 1.

Class 7 Maths Chapter 13 Notes

Decimal Number System
10,000 = 10⁴
1000 = 10³
100 = 10²
10 = 10¹
1 = 10⁰
We can write the expansion of a number using powers of 10 in the exponent form.

Exponents And Powers Notes Class 7

Expressing Large Numbers in the Standard Form
Large numbers can be expressed conveniently using exponents. Such a number is said to be in standard form if it can be expressed as k × 10^m, where 1 ≤, k < 10 and m is a natural number.

Note that, one less than the digit count (number of digits) to the left of the decimal point in a given number, is the exponent of 10 in the standard form.

For any rational number a and positive integer n, we define aⁿ as a × a × a × …… × a (n times). aⁿ is known as the nth power of a and is read as ‘a raised to the power n’. The rational a is called the base and n is called the exponent or power.
e.g. 10,000 = 10 × 10 × 10 × 10 = 10⁴.
10 is the base and 4 is the exponent.

Multiplying Powers with the Same Base: If a is any non-zero integer and whole numbers are m and n, then a^m × aⁿ = a^m+n
e.g. 2⁴ × 2²
a = 2, m = 4, n = 2
2⁴ × 2² = 2⁴⁺² = 2⁶

Dividing Powers with the Same Base: If a is any non-zero integer and m, n are the whole number, then a^m ÷ aⁿ = a^m-n
e.g. 2⁴ ÷ 2²
a = 2, m = 4, n = 2
2⁴ ÷ 2² = 2^4-2 = 2²

Taking Power of a Power: If a is any non-zero integer and m, n are whole numbers, (a^m)ⁿ = a^mn
e.g. (6²)⁴
a = 6, m = 2, n = 4
(6²)⁴ = (6)^2×4 = 6⁸.

Multiplying Powers with the Same Exponents: If a, b are two non-zero integers and m is any whole number, then
a^m × bⁿ = (a × b)^m
e.g. 2³ × 3³
a = 2, b = 3, m = 3
2³ × 3³ = (2 × 3)³ = 6³.

Dividing Powers with the Same Exponents: If a, b are two non-zero integers and m is a whole number, then

Numbers with Exponent Zero: If a be any non-zero integer, then, a⁰ = 1

Numbers with Negative Exponent: If a is any non-zero integer, then a^-1 = \(\frac { 1 }{ a }\)
e.g. 2^-5 = \(\frac { 1 }{ { 2 }^{ 5 } }\)

In decimal number system, the exponents of 10 start from a maximum value and go on decreasing from the left to right upto 0.
e.g. 45672 = 4 × 10000 + 5 × 1000 + 6 × 100 + 7 × 10 + 2 × 1
= 4 × 10⁴ + 5 × 10³ + 6 × 10² + 7 × 10¹ + 2 × 10⁰
It is called expanded form of a number.

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.
e.g. 56782 = 5.6782 × 10000 = 5.6782 × 10⁴.
It is the standard form of 56782.

Class 7 Maths Notes students can refer to the Symmetry Class 7 Notes Maths Chapter 14 https://www.cbselabs.com/symmetry-class-7-notes/ Pdf here. They can also access the CBSE Class 7 Symmetry Chapter 14 Notes while gearing up for their Board exams.

CBSE Class 7 Maths Notes Chapter 14 Symmetry

Symmetry Class 7 Notes Chapter 14

Lines of Symmetry for Regular Polygons
Regular polygons have equal sides and equal angles. They have multiple (i.e., more than one) lines of symmetry. Each regular polygon has as many lines of symmetry as it has sides.

Regular Polygon	Regular Hexagon	Regular Pentagon	Square	Equilateral Triangle
Number of Lines of Symmetry	6	5	4	3

Symmetry Notes Class 7 Chapter 14

Rotational Symmetry
Rotation, like the movement of the hands of a clock, is called a clockwise rotation; otherwise, it is said to be anticlockwise.

When an object rotates, its shape and size do not change. The rotation turns an object about a fixed point. This fixed point is called the centre of rotation. The angle by which the object rotates is called the angle of rotation.

A half-turn means rotation by 180°; a quarter-turn means rotation by 90°.

Rotation may be clockwise or anticlockwise.

If, after a rotation, an object looks exactly the same, we say that it has rotational symmetry.

In a complete turn (of 360°), the number of times an object looks exactly the same is called the order of rotational symmetry. For example, the order of symmetry of a square is 4 while, for an equilateral triangle, it is 3.

Class 7 Maths Symmetry Notes Chapter 14

Line Symmetry and Rotational Symmetry
Some shapes have only one line of symmetry, like the letter E;

Some have only rotational symmetry, like the letter S;

and some have both symmetries like the letter H.

The study of symmetry is important because of its frequent use in day-to-day life and more because of the beautiful designs it can provide us.

A figure is said to be symmetrical about a line l if it is identical on either side of l. In the adjoining figure, / is the line of symmetry or axis of symmetry.

Regular polygons have equal sides and equal angles. They have multiple (i.e. more than one) lines of symmetry.

Each regular polygon has as many lines of symmetry as it has sides.

Lines of Symmetry of some Irregular Polygons.

Each of the following capital letters of the English alphabet is symmetrical about the dotted line or lines as shown:

A figure is said to have rotational symmetry if, after a rotation, an object looks exactly the same.

The fixed point, about which the rotation turns an object (not changing its shape and size) is called centre of rotation.

In a complete turn (of 360°), the number of times an object looks exactly the same is called the order of rotational symmetry.

Class 7 Maths Notes students can refer to the Practical Geometry Class 7 Notes Maths Chapter 10 https://www.cbselabs.com/practical-geometry-class-7-notes/ Pdf here. They can also access the CBSE Class 7 Practical Geometry Chapter 10 Notes while gearing up for their Board exams.

CBSE Class 7 Maths Notes Chapter 10 Practical Geometry

Practical Geometry Class 7 Notes

Construction of Triangles

1. Properties of triangles

• The exterior angle of a triangle is equal in measure to the sum of interior opposite angles.
• The total measure of the three angles of a triangle is 180°.
• Sum of the lengths of any two sides of a triangle is greater than the length of the third side.
• In any right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

2. Essential measurements for the construction of a triangle
A triangle can be drawn if any one of the following sets of measurements is given:

• Three sides: SSS
• Two sides and the angle between them: SAS
• Two angles and the side between them: ASA
• The hypotenuse and a leg in the case of a right-angled triangle: RHS

Construction of a line parallels to a given line, through a point not on the line.

Steps of Construction

Step 1: Take a line ‘l’ and a point ‘A’ outside ‘l’

Step 2: Take any Point B on l and join B to A.

Step 3: With B as centre and a convenient radius, draw an arc cutting l at C and BA at D.

Step 4: Now with A as centre and the same radius as in Step 3 draw an arc EF cutting AB at G.

Step 5: Place the pointed tip of the compasses at C and adjust the opening so that the pencil tip at D.

Step 6: With the same opening as in Step 5 and with G as centre, draw an arc cutting the arc EF at H.

Step 7: Now, join AH to draw a line ‘m’.

Construction of a triangle when the lengths of its three sides are known (SSS Criterion).

Steps of Construction
Step 1: First, we draw a rough sketch with a given measure.

Step 2: Draw a line segment BC of length 6 cm.

Step 3: From B, point A is at a distance of 5 cm. So with B as centre, draw an arc of radius 5 cm.

Step 4: From C, point A is at a distance of 7 cm. So, with C as centre, draw an arc of radius 7 cm.

Step 5: A has to be on both the arcs drawn. So, it is the point of intersection of arc.
Mark the point of intersection of arcs as A. Join AB and AC. ΔABC is now ready.

Constructing a triangle when the lengths of two sides and the measure of the angle between them are known. (SAS Criterion)

Steps of Construction
Step 1: First we draw a rough sketch with given measures.

Step 2: Draw a line segment QR of length 5.5 cm.

Step 3: At Q, draw QX making 60° with QR.

Step 4: With Q as centre, draw an arc of radius 3 cm. It cuts QX at the point E

Step 5: Join PR. ΔPQR is now obtained.

Constructing a triangle when the measures of two of its angles and the length of the side included between them is given (ASA Criterion)

Steps of Construction
Step 1: Before actual construction, we draw a rough sketch with measures marked on it.

Step 2: Draw XY of length 6 cm.

Step 3: At X, draw a ray XP making an angle of 30° with XY. By the given condition Z must be somewhere on the XP.

Step 4: At Y, draw a ray YQ making an angle of 100° with YX. By the given condition, Z must be on the ray YQ also.

Step 5: Z has to lie on both the rays XP and YQ. So, the point of intersection of two rays is Z.

ΔXYZ is now completed.

Constructing a Right-Angled Triangle when the length of one leg and its hypotenuse are given (RHS Criterion).

Steps of Construction
Step 1: Draw a rough sketch and mask the measures. Remember to mark the right angle.

Step 2: Draw MN of length 3 cm.

Step 3: At M, draw MX ⊥ MN.

Step 4: With N as centre, draw an arc of radius 5 cm.

Step 5: L has to be on the perpendicular line MX as well as on the arc drawn with centre N. Therefore, L is the meeting point of these two ΔLMN is now obtained.

CBSE Class 7 Maths Notes Chapter 8 Comparing Quantities Pdf free download is part of Class 7 Maths Notes for Quick Revision. Here we have given NCERT Class 7 Maths Notes Chapter 8 Comparing Quantities. https://www.cbselabs.com/comparing-quantities-class-7-notes/

CBSE Class 7 Maths Notes Chapter 8 Comparing Quantities

Comparing Quantities Class 7 Notes Chapter 8

To compare quantities, there are multiple methods, such as ratio and proportion, percentage, profit and loss, and simple interest.

The ratio of two quantities of the same kind and in the same unit is the fraction that one quantity is of the other.

The ratio a is to b as is the fraction \(\frac { a }{ b }\), and it is written as a : b.

In the ratio a : b, we call a as the first term or antecedent and b the second term or consequent.

To compare different ratios, firstly convert fractions into like fractions. If like fractions are equal, then the given ratios are said to be equivalent.
e.g. To check 1 : 2 and 2 : 3 are equivalent.

Therefore, the ratio 1 : 2 is not equivalent to the ratio 2 : 3.

To compare two quantities, units must be the same.

If the two ratios are equal, the four quantities are called in proportion.
a : b = c : d ⇒ a : b :: c : d.

Fractions are converted to percentages by multiply the fraction by 100 and write % sign
e.g. \(\frac { 1 }{ 4 }\) = \(\frac { 1 }{ 4 }\) × 100 = 25%

Decimals are converted to percentages by multiply the decimal number by 100 and shift the decimal point two places to the right side and write % sign.
e.g. 2.42 × 100 = 242 %

Profit = SP – CP [∵ SP > CP]
SP = Selling Price
CP = Cost Price

Loss = CP – SP [∵ CP > SP]
CP = Cost Price
SP = Selling Price

Profit % = \(\frac { Profit }{ CP }\) × 100
Loss % = \(\frac { Loss }{ CP }\) × 100

Class 7 Comparing Quantities Notes Chapter 8

Equivalent Ratios
Different ratios are compared with each other to know whether they are equivalent or not. For this, we write the ratios in the form of fractions and then compare them by converting them into like fractions. If these like fractions are equal, we say that the given ratios are equivalent. Equivalent ratios are very important. Two ratios are said to be equivalent if when converted into like fractions, they are equal.

Unitary Method
In the unitary method, we first find the value of one unit and then the value of the required number of units.

Percentage-Another way of Comparing Quantities
Percentages are numerators of fractions with denominator 100. They are used for comparisons.

Meaning of Percentage
Percent means ‘per hundred’. It is represented by the symbol % and means hundredths too.
Thus, 1% means 1 out of hundred or one-hundredths. It can be written as:
1% = \(\frac { 1 }{ 100 }\) = 0.01

Class 7 Maths Chapter 8 Notes

Converting Fractional Numbers to Percentage
Fractional numbers can have different denominators. To compare fractional numbers we need a common denominator and it is more convenient to compare if the denominator is 100. So, we convert the fraction to percentages.
Percentages related to proper fractions are less than 100 whereas percentages related to improper fractions are more than 100.

Converting Decimals to Percentage
We multiply the decimal by 100 and affix percentage symbol.

Ratios to Percents
Sometimes, parts are given to us in the form of ratios and we need to convert those to percentage.

Increase or Decrease as Percent
There are times when we need to know what the increase in a certain quantity or decrease in it is as percent.
For example, if the population of a state is increased from 5,50,000 to 6,05,000, this could more clearly be understood if written as:
The population is increased by 10%.

Prices Related to an Item on Buying and Selling
The buying price of an item is known as its Cost Price (CP).
The price at which we sell an item is known as its Selling Price (SP).

Comparing Quantities Notes Class 7 Chapter 8

Profit or Loss as a Percentage
Cost Price: The buying price of an item is known as its cost price written in short as CP.
Selling Price: The price at which we sell an item is known as the selling price or in short SP.
Naturally, it is better if we sell the item at a higher price than our buying price.
Profit or Loss: We can decide whether the sale was profitable or not depending on the CP and SP.

If CP < SP then we have gained some amount, that is, we made a profit, profit = SP – CP
If CP = SP then we are in a no profit no loss situation
If CP > SP then we have lost some amount, Loss = CP – SP.
The profit or loss we find can be converted to a percentage. It is always calculated on the CP.
Note. If we are given any two of the three quantities related to price, that is, CP, SP, and Profit or Loss percent, we can find the third.

Charge has given on Borrowed Money or Simple Interest
Principal: The money borrowed is known as sum borrowed or principal.

Interest: We have to pay some extra money (or charge) to the bank for the money being used by us for some time. This is known as interest.

Amount: We can find the amount we have to pay at the end of the year by adding the above two. That is.
Amount = Principal + Interest.

Note: Interest is generally given in per cents for a period of one year. It is written as x percent per year or per annum or in short as x% p.a. (say 10 percent per year)
10% p.a. means on every ₹ 100 borrowed, ₹ 10 is the interest we have to pay for one year.

We hope the given CBSE Class 7 Maths Notes Chapter 8 Comparing Quantities Pdf free download will help you. If you have any query regarding NCERT Class 7 Maths Notes Chapter 8 Comparing Quantities, drop a comment below and we will get back to you at the earliest.

CBSE Class 7 Maths Notes Chapter 7 Congruence of Triangles Pdf free download is part of Class 7 Maths Notes for Quick Revision. Here we have given NCERT Class 7 Maths Notes Chapter 7 Congruence of Triangles. https://www.cbselabs.com/congruence-of-triangles-class-7-notes/

CBSE Class 7 Maths Notes Chapter 7 Congruence of Triangles

Congruence Of Triangles Class 7 Notes Chapter 7

If two figures have exactly the same shape and size, then they are said to be congruent.
For congruence, we use the symbol ‘=’

Two plane figures are congruent, if each when superposed on the other covers it exactly.
e.g. F₁ and F₂ are congruent if the trace copy of F₁ fits exactly on that F₂. We can write this as F₁ = F₂

Two line segments, \(\bar { AB }\) and \(\bar { CD }\) are congruent if they have equal lengths. We can write this as \(\bar { AB }\) = \(\bar { CD }\). However, it is common to write it as \(\bar { AB }\) = \(\bar { CD }\).

Two angles ∠ABC and ∠PQR, are congruent if their measures are equal. We can write this as ∠ABC = ∠PQR or m∠ABC = m∠PQR. Also, it is common to write it as ∠ABC = ∠PQR.

Class 7 Congruence Of Triangles Notes Chapter 7

SSS Congruence of two triangles: Two triangles are congruent if the three sides of the one are equal to the three corresponding sides of the other.

where, AB = PQ, AC = PR, BC = QR

SAS Congruence of two triangles: Two triangles are congruent if two sides and the angle included between them in one of the triangles are equal to the corresponding sides and the angle included between them of the other triangle.

where, AB = PQ, AC = PR, ∠BAC = ∠QPR.

Class 7 Maths Chapter 7 Notes

ASA congruence of two triangles: Two triangles are congruent if two angles and the side included between them in one of the triangles are equal to the corresponding angles and the side included between them of the other triangle.

where, BC = QR, ∠ABC = ∠PQR, ∠ACB = ∠PRQ.

RHS Congruence of two right-angled triangles: Two right-angled triangles are congruent if the hypotenuse and a leg of one of the triangles are equal to the hypotenuse and the corresponding leg of the other triangle.

where, AC = PR, BC = QR, ∠ABC = ∠PQR = 90°.

Congruence Of Triangle Class 7 Notes Chapter 7

Congruence of Plane Figures
Two figures F₁ and F₂ are said to be congruent if they cover each other completely.
In this case, we write F₁ = F₂.

Congruence Among Line Segments
If two line segments have the same (i.e., equal) length, they are congruent. Conversely, if two line segments are congruent, they have the same length.
Line segments are congruent ⇔ their lengths are the same
If line segment AB is congruent to line segment CD, then we write AB = CD
Sometimes we also write AB = CD
and simply say that the line segments AB and CD are equal.

Congruence Of Triangles Class 7 Pdf Notes Chapter 7

Congruence of Angles
If two angles have the same measure, they are congruent.
Conversely, if two angles are congruent, their measures are the same.
Angles congruent ⇔ Angle measures same
If ∠ABC is congruent to ∠XYZ, then we write ∠ABC = ∠XYZ  or m∠ABC = m∠XYZ
where m stands for the measure.

In the above case, the measure is 60°.

Congruence of Triangles
“Two triangles are congruent if they are copies of each other and when superposed, they cover each other exactly.”

∆ABC and ∆PQR have the same size and shape. They are congruent. So we would express this as ∆ABC = ∆PQR.
This means that, when we place ∆PQR on ∆ABC, P falls on A, Q falls on B and R falls on C, also \(\bar { PQ }\) falls along \(\bar { AB }\), \(\bar { QR }\) falls along \(\bar { BC }\) and \(\bar { PR }\) falls along \(\bar { AC }\).

If under a given correspondence, two triangles are congruent, then their corresponding parts (i.e. angles and sides) that match one another are equal. Thus in these two congruent triangles, we have:
Corresponding vertices: A and P, B and Q; C and R.
Corresponding sides: \(\bar { AB }\) and \(\bar { PQ }\), \(\bar { BC }\) and \(\bar { QR }\); \(\bar { AC }\) and \(\bar { PR }\).
Corresponding angles: ∠A and ∠P, ∠B and ∠Q; ∠C and ∠R.

While talking about the congruence of triangles, not only the measures of angles and lengths of sides matter, but also the matching of vertices. In the above case, the correspondence is A ↔ P, B ↔ Q, C ↔ R
We may write this as ABC ↔ PQR

Class 7 Chapter 7 Maths Notes

Criteria for Congruence of Triangles
SSS Congruence Criterion
If under a given correspondence, the three sides of one triangle are respectively equal to the three sides of another triangle, then the triangles are congruent.

SAS Congruence Criterion
If under a corresponding two sides and the angle included between them of a triangle are respectively equal to two sides and the angle included between them of another triangle, then the triangles are congruent.

ASA Congruence Criterion
If under a correspondence two angles and the included side of a triangle are equal to two angles and the included side of another triangle, then the triangles are congruent.

Note: If two angles of a triangle are given, we can always find the third angle of the triangle. So, if, two angles and one side of one triangle are equal to the two angles and the corresponding side of another triangle, then the third angle of the two triangles shall also be equal (as the sum of the three angles of a triangle is 180°), so we may convert it into “two angles and the included side” form of congruence and then apply the ASA congruence rule.

Congruence Of Triangles Class 7 Chapter 7

Congruence Among Right-Angled Triangles
RHS Congruence Criterion
If under a correspondence, the hypotenuse and one side of a right-angled triangle are respectively equal to the hypotenuse and one side of another right-angled triangle, then the triangles are congruent.
We call this “RHS” congruence because R stands for Right-angle, H stands for Hypotenuse and S stands for Side.

We hope the given CBSE Class 7 Maths Notes Chapter 7 Congruence of Triangles Pdf free download will help you. If you have any query regarding NCERT Class 7 Maths Notes Chapter 7 Congruence of Triangles, drop a comment below and we will get back to you at the earliest.