Chapter - 1 : Integers Let’s Revise Counting numbers (natural numbers) are 1, 2, 3, 4, ........ Whole numbers are 0, 1, 2, 3, ........ ........., – 4, – 3, – 2, – 1, 0, 1, 2, 3, ......., this sequence is called the set of integers. 0 is called the additive Identity, as 0 + a = a for any number a. When two positive integers are added, we get a positive integer. e.g. 18 + 2 = 20. When two negative integers are added, we get a negative integer. e.g. – 6 + (– 2) = – 8 The additive inverse of any integer x is – x, and the additive inverse of (– x) is x. If a and b are integers, then (a + b), (a – b) and (a × b) are also integers. If a and b are integers and a + b = b + a, then addition is commutative for integers. In the same way multiplication is commutative for integers , i.e., a × b = b × a. If there are three integers a, b and c, then (a) addition is associative, i.e., (a + b) + c = a + (b + c) (b) multiplication is associative, i.e., (a × b) × c = a × (b × c) (c) multiplication is distributed over addition and subtraction, i.e., a × (b + c) = (a × b) + (a × c) and a × (b – c) = (a × b) – (a × c) 1 is the multiplicative identity, a.1 = a = 1.a for any number a. For integers, multiplication of same signs is positive and multiplication of opposite signs is negative. If 0 is multiplied to any integer, product is always 0, a.0 = 0 for any number a. Integer 0 = not defined (∞), but 0 Integer = 0, for non-zero integers. Product of a positive integer and a negative integer is a negative integer, i.e., a × (–b) = – ab, where a and b are integers. Product of two negative integers is a positive integer, i.e., (–a) × (–b) = ab, where a and b are integers. Product of even number of negative integers is positive, where as the product of odd number of negative integers is negative, i.e., ( ) ( ) ... ( ) - - - a b p × × × even number 2m times = a × b × .... × p and ( ) ( ) ... ( ) ( ) - - - a b q × × × + odd number m times 2 1 = –(a × b × ... × q), where a, b, ..., p, q and m are integers. When a positive integer is divided by a negative integer or vice-versa and the quotient obtained is an integer, then it is a negative integer, i.e., a ÷ (–b) = (–a) ÷ b = – a b , where a and b are positive integers and a b is an integer. When a negative integer is divided by another negative integer to give an integer, then it gives a positive integer, i.e., (–a) ÷ (–b) = a b , where a and b are positive integers and a b is also an integer. For any integer a, a ÷ 1 = a and a ÷ 0 is not defined. Know the Terms Natural numbers : The counting numbers 1, 2, 3, 4, .... are called natural numbers. Whole numbers : The collection of ‘0’ and all natural numbers are called whole numbers. Integers : The collection of numbers, which contain whole numbers and negative numbers. e.g., –5, – 2, 0, 1, 3, 5, etc.
Transcript
Chapter - 1 : Integers
Let’s Revise Counting numbers (natural numbers) are 1, 2, 3, 4, ........ Whole numbers are 0, 1, 2, 3, ........ ........., – 4, – 3, – 2, – 1, 0, 1, 2, 3, ......., this sequence is called the set of integers. 0 is called the additive Identity, as 0 + a = a for any number a. When two positive integers are added, we get a positive integer. e.g. 18 + 2 = 20. When two negative integers are added, we get a negative integer. e.g. – 6 + (– 2) = – 8 The additive inverse of any integer x is – x, and the additive inverse of (– x) is x. If a and b are integers, then (a + b), (a – b) and (a × b) are also integers. If a and b are integers and a + b = b + a, then addition is commutative for integers. In the same way multiplication is commutative for integers , i.e., a × b = b × a. If there are three integers a, b and c, then (a) addition is associative, i.e., (a + b) + c = a + (b + c) (b) multiplication is associative, i.e., (a × b) × c = a × (b × c) (c) multiplication is distributed over addition and subtraction, i.e., a × (b + c) = (a × b) + (a × c) and a × (b – c)
= (a × b) – (a × c) 1 is the multiplicative identity, a.1 = a = 1.a for any number a. For integers, multiplication of same signs is positive and multiplication of opposite signs is negative. If 0 is multiplied to any integer, product is always 0, a.0 = 0 for any number a.
Integer
0 = not defined (∞), but
0Integer = 0, for non-zero integers.
Product of a positive integer and a negative integer is a negative integer, i.e., a × (–b) = – ab, where a and b are integers.
Product of two negative integers is a positive integer, i.e., (–a) × (–b) = ab, where a and b are integers.
Product of even number of negative integers is positive, where as the product of odd number of negative integers is negative, i.e.,
( ) ( ) ... ( )- - -a b p× × ×even number 2m times
= a × b × .... × p and
( ) ( ) ... ( )
( )- - -a b q× × ×
+odd number m times2 1 = –(a × b × ... × q), where a, b, ..., p, q and m are integers.
When a positive integer is divided by a negative integer or vice-versa and the quotient obtained is an integer, then it is a negative integer, i.e.,
a ÷ (–b) = (–a) ÷ b = –
ab
, where a and b are positive integers and
ab
is an integer.
When a negative integer is divided by another negative integer to give an integer, then it gives a positive integer,
i.e., (–a) ÷ (–b) =
ab
, where a and b are positive integers and
ab
is also an integer.
For any integer a, a ÷ 1 = a and a ÷ 0 is not defined.
Know the Terms Natural numbers : The counting numbers 1, 2, 3, 4, .... are called natural numbers. Whole numbers : The collection of ‘0’ and all natural numbers are called whole numbers. Integers : The collection of numbers, which contain whole numbers and negative numbers. e.g., –5, – 2, 0, 1, 3, 5,
etc.
2 ] Oswaal CBSE Chapterwise Quick Review, MATHEMATICS, Class-VII Number line : On a number line, when we
(a) add a positive integer, we move to the right.
(b) add a negative integer, we move to the left.
(c) subtract a positive integer, we move to the left.
(d) subtract a negative integer, we move to the right.
Some integers are marked on the number line as shown below :
–6–5–4–3 –2 –1 0 1 2 3 4 5 6 7
The ascending order of these numbers is –5, –1, 3. Integers of Number line :
Ex. An elevator descends into a mine shaft at the rate of 6 m/min. If the descent starts from 10 m above the ground level, how long will it take to reach –350 m ?
Sol. Step I : Starting position of mine shaft is 10 m above the ground level.
If travels towards the ground i.e., opposite to the initial displacement above the ground.
Step II : Distance travelled below the ground = 350 m
Step III : Total distance travelled = 10 – (– 350) m
= 360 m
Step IV : Time taken to travel 1 m = 1
6min
Step V : Time taken to travel 360 m =
1
6 × 360 min
= 60 min
= 1 hour
Chapter - 2 : Fractions and Decimals
Let’s Revise The study of fractions included proper, improper and mixed fractions as well as their addition and subtraction. Decimal included their comparison, their representation on the number line and their addition and subtraction.
Fraction : The number of the form
ab
, where a and b are natural numbers, is known as fraction.
e.g.,
35
is a fraction, where 3 is numerator and 5 is denominator.
Various types of fractions : l Decimal fraction : A fraction whose denominator is any of the numbers 10, 100, 1000, etc., is called a decimal
fraction.
e.g.,
210
4100
131000
, , etc., are all decimal fractions.
l Vulgar fraction : A fraction whose denominator is a whole number, other than 10, 100, 1000, etc., is called a vulgar fraction.
e.g., 29
413
1320
27109
, , , , etc., are all vulgar fractions. It is also known as simple or common fraction.
Oswaal CBSE Chapterwise Quick Review, MATHEMATICS, Class-VII [ 3 l Proper fraction : A fraction whose numerator is less than its denominator is called a proper fraction.
e.g., 37
511
2340
73100
, , , , etc., are all proper fractions.
l Improper fraction : A fraction whose numerator is more than or equal to its denominator is called an improper fraction.
e.g.,
117
2512
4136
5353
, , , , etc., are all improper fractions.
l Mixed fraction : A number which can be expressed as the sum of a natural number and a proper fraction is called a mixed fraction.
e.g.,
1
34
457
79
1312
625
, , , , etc., are all mixed fractions.
l Equivalent fractions : A given fraction and the fraction obtained by multiplying (or dividing) its numerator and denominator by the same non-zero number are called equivalent fractions.
lLike fractions : Fractions having the same denominator but different numerators are called like fractions.
e.g., 5
136
13713
, , , etc., are like fractions.
lUnlike fractions : Fractions having different denominators are called unlike fractions.
e.g., 37
59
1113
, , , etc. are unlike fractions.
Method of comparing more than two fractions Step I : Find LCM of the denominators of the given fractions. Let it be m. Step II : Convert all the given fractions into like fractions, each having m as denominator. Step III : Now, if we compare any two of these like fractions, then the one having larger numerator is larger. Properties of addition of fractions : (a) Addition of fractions is associative
i.e.,
ab
cd
ef
ab
cd
ef
+
+ = + +
e.g.,
23
45
76
23
45
76
+
+ = + +
LHS =
23
45
76
10 1215
76
2215
76
44 3530
7930
+
+ =
++ = + =
+=
RHS =
23
45
76
23
24 3530
23
5930
20 5930
7930
+ +
= +
+
= + =
+=
∴ LHS = RHS
(b) Addition of fractions is commutative i.e.,
ab
cd
cd
ab
+
= +
.
e.g.,
23
45
45
23
+
= +
Rule 2 : For addition of two unlike fractions, first change them to like fractions and then add them as given in Rule 1.
e.g.,
43
34
4 43 4
3 34 3
+ =××
+××
=
1612
912
16 912
2512
+ =+
=
Addition of fractions Rule 1 : For adding two like fractions, the numerators are added and the denominator remains the same.
e.g., (a)
29
59
2 59
79
+ =+
=
(b)
815
615
8 615
1415
+ =+
=
Note : The subtraction of fractions can be done in same way as addition. Reciprocal of fractions : Two fractions are said to be the reciprocal of each other, if their product is 1.
e.g.,
49
and
94
are the reciprocal of each other, since
49
94
×
= 1.
In general, if ab
is a fraction, then its reciprocal is
ba
.
4 ] Oswaal CBSE Chapterwise Quick Review, MATHEMATICS, Class-VII Division of whole number by a fraction : To divide a whole number by any fraction, multiply that whole number
by the reciprocal of that fraction.
e.g., 3 ÷
14
341
= × = 12
Division of a whole number by a mixed fraction : To divide a whole number by a mixed fraction, firstly convert the mixed fraction into improper fraction and then solve it.
e.g., 4 ÷ 2
25
= 4 ÷
125
= 4 ×
512
=
2012
Division of a fraction by a whole number : To divide a fractional number by a whole number, multiply the fractional number by the reciprocal of the whole number.
Division of a mixed fraction by a whole number : To divide a mixed fraction by a whole number, convert the mixed fraction into improper fraction and then solve it.
e.g.,
2
23
583
583
15
815
÷ = ÷ = × =
Division of a fraction by another fraction : Rule : To divide a fraction by another fraction, the first fraction is multiplied by the reciprocal of the second
fraction.
Thus,
ab
cd
÷
=
ab
dc
×
e.g.,
23
45
23
54
1012
÷
= × =
Decimals : The numbers expressed in decimal forms are called decimals. e.g., 6.8, 16.73, 7.364, 0.053, etc., are all decimal numbers. A decimal has two parts, namely (a) whole number part
and (b) decimal part. Multiplication of fractions : l When we multiply a whole number with a proper or improper fraction, we multiply the whole number with
numerator of fraction keeping the denominator same.
e.g., 4 ×
23
=
4 23×
=
83
and 7 ×
53
=
7 53×
=
353
l Two fractions are multiplied by multiplying their numerators and denominators separately and writing the product as
Product of numeratorsProduct of denominators
e.g.,
25
34
2 35 4
620
× =××
=
l A fraction acts as an operator ‘of ’.
e.g.,
13
of 3 is
13
× 3 = 1.
l The product of two proper fractions is less than each of the fractions,
e.g.,
12
13
16
× = and
16
is less than both
12
and
13
.
l The product of a proper and an improper fraction is less than the improper fraction and greater than the
proper fraction. e.g.,
12
32
34
× = and
34
is less than
32
but greater than
12
.
l The product of two improper fractions is greater than the two fractions.
e.g.,
32
74
218
× = and
218
is greater than both
32
and
74
.
l To multiply a whole number by a decimal number, ignore the decimals and multiply the two numbers, then count the number of digits to the right of decimal point in the original decimal number and insert the decimal from right to left in the answer by the same count.
e.g., (a) 3 × 0.2 = 0.6 (b) 3 × 0.4 = 1.2
Oswaal CBSE Chapterwise Quick Review, MATHEMATICS, Class-VII [ 5 To divide a decimal number by a whole number, convert the decimal number into a fraction, then take the
reciprocal of the division and multiply the reciprocal by the fraction.
e.g.,
0 143
14100
13
14300
·= × =
While multiplying two decimal numbers, first multiply them as whole numbers. Count the number of digits to the right of the decimal point in both the decimal numbers. Add the number of digits counted. Put the decimal point in the product by counting the number of digits equal to sum obtained from its rightmost place.
e.g., 1.2 × 1.24 = 1.488. To multiply a decimal number by 10, 100 or 1000, we move the decimal point in the number to the right by as
many places as many zeros (0) are the right of one e.g., 1.33 × 10 = 13.3. To divide a decimal number by a natural number, we first take the decimal number as natural number and divide
by the given natural number. Then place the decimal point in the quotient as in the decimal number.
e.g.,
1 24.
= 0.3
To divide a decimal number by 10, 100 or 1000, shift the decimal point in the decimal number to the left by as many places as there are zeros over 1, to get the quotient.
e.g.,
1 34100.
= 0.0134
While dividing one decimal number by another, first shift the decimal points to the right by equal number of places in both, to convert the divisor to a natural number and then divide.
e.g.,
1 441 2
14 412
..
.=
=
1.2
GREENBOARD ?How it is done on
Q. Evaluate :
32
÷
914
Sol. : Step-I : Division is opposite process of multiplication.
Step-II : we have, 32
÷
914
Step-III : ∴
32
×
149
=
73
Chapter - 3 : Data Handling
Let’s Revise The information collected in the form of numbers is called Data. Data is organised and represented graphically so
that it becomes easy to understand and interpret. The collection, recording and presentation of data help us in organising our experiences and draw inferences
from them. Average is a number that represents or shows the central tendency of a group of observations or data. Collecting and organisation of data : Collecting data and further organising it in a particular manner, makes it
easier for us to understand and interpret data. Also, before collecting the data, one needs to known the purpose for which data will be used.
Representative value : Representative value or central value is a measure of central tendency of the group of data. In our day-to-day life, we come across many statements that involve the term ‘average’. Average is a number indicating the representative or central value of a group of observations.
e.g., The average age of students in a class is 12 year, the average temperature at this time of the year is 40°C. Similarly, different forms of data need different forms of representative or central value to describe them. The most common of them are arithmetic mean, mode and median.
Arithmetic mean : The most common representative value of a group of data is the arithmetic mean or mean.
6 ] Oswaal CBSE Chapterwise Quick Review, MATHEMATICS, Class-VII The average or arithmetic mean (AM) or simply mean is defined as follows :
Arithmetic Mean =
Sum of all observationsNumber of observations
Mean always lies between the greatest and smallest observation of the data. Range : It is the difference between the highest and the lowest observation of the data. It gives us an idea of the
spread of observations. i.e., Range = Highest observation – Lowest observation Mode : Mode of a set of observation is the observation that occurs, the most often. e.g. The mode of set of numbers
1, 1, 2, 4, 3, 2, 1, 2, 2, 4 is 2 because it occurs more frequently than other observations. In order to find the mode of larger data, tabulation of data is done. Tabulation can begin by putting tally marks and then, finding the frequency.
The following are the steps to calculate mode : Step 1 : Arrange the data in ascending order. 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 which lies in the middle of the data (when arranged in an increasing or
decreasing order) with half of the observation above it and other half below it, e.g., For finding the median of the data 24, 36, 46, 17, 18, 25, 35. Firstly data is to be arranged in ascending order i.e., 17, 18, 24, 25, 35, 36, 46. Since, median is the middle observation, therefore 25 is the median.
Note (i) If the data has odd number of items, then the median is the middle number. (ii) If the data has an even number of items, then the median is the mean of two middle numbers. Bar graph : Bar graph is an visual representation of data. It is formed by using bars of uniform widths. One can
look at the bar graph and make deductions about the data. Also, one can get information based on these bar graphs. e.g., Mode of the data is the longest bar, if the bar represents the frequency.
Choosing a scale : In order to draw a bar graph, an appropriate scale needs to be chosen. e.g., In a bar graph, where numbers in units are to be shown, the graph represents one unit length for one observation and if it has to show numbers in tens or hundred, one unit length can represent 10 or 100 observations.
Double bar graph : A double bar graph can be drawn to compare two sets of observations. e.g., If we have two collections of data giving average daily hours of sunshine in two cities for twelve months. Then, a double bar graph will be useful in finding information e.g., In a particular month which city has more sunshine hours.
The situation that may or may not happen, have a chance of happening.
The probability of an event =
Number of favourable outcomesTotal number of outcomes in thhe experiment
The probability of an event which is certain to happen is ‘1’.
The probability of an event which is impossible to happen is ‘0‘.
GREENBOARD ?How it is done on
Q. Given below are heights of 15 boys of a class meas-ured in cm :
Find (a) The height of the tallest boy. (b) The height of the shortest boy. (c) The range of the given data. (d) The median height of the boys.Sol. : Step-I : First we arrange the given data in ascend-
Let’s Revise Linear equation : An equation is a statement of equality, which contains one or more unknown quantities or
variables.OR
‘An equation involving only a linear polynomial is called a linear equation.’ Note : (i) A linear equation remains the same when the expression in the left and right are interchanged. (ii) In an equation, there is always an equality sign. Solution of a linear equation : The value of the variable, which makes the equation a true statement is called the
solution or root of a linear equation. e.g., 5x – 12 = – 2 is a equation. If x = 2, then LHS = 5x – 12 = 5 × 2 – 12 = 10 – 12 = – 2 LHS = RHS Trial and error method : l In trial and error method, putting the different values for the variables 0, 1, 2, 3,... one by one and then find
the corresponding values of LHS and RHS. l The value of variable, for which LHS = RHS, will be the solution of the linear equation. Rules for solving an equation : (a) The same quantity can be added to both sides of an equation without changing the equality. (b) The same quantity can be subtracted from both sides of an equation without changing the equality. (c) Both sides of an equation may be multiplied by the same non-zero number without changing the equality. (d) Both sides of an equation may be divided by the same non-zero number without changing the equality. Transposition : Any term of an equation may be taken from one side to the other with a change in its sign. This
does not affect the equality of the statement and this process is called transposition.
GREENBOARD ?How it is done on
Q.
If 45 is added to half a number, the result is triple the number. Find the number.
Sol. Step-I : Let the number be x.
Step-II : Half the number = x
2
Step-III : Add 45 to half the number i.e., x
2 + 45
Step-IV : According to the question,
3x = x
2 + 45
Step-V : Solve the equation
3x –
x
2 = 45
⇒
5x
2 = 45
⇒ x = 18 Step-VI : Hence, the number is 18.
Chapter - 5 : Lines and Angles
Let’s ReviseLine segment : A part of a line with two end points is called line segment
and it is denoted by AB . Ray : It is a part of a line with one end point and its other end can be
extended further. It is denoted by AB� ���
.
Line : It can be extended from both sides (left and right) and it is denoted by AB� ���
.
A B
A B
A B
A B C
8 ] Oswaal CBSE Chapterwise Quick Review, MATHEMATICS, Class-VII Collinear points : Three or more points are said to be collinear, if a single
straight line passes through them.
Here, A, B and C are collinear points. Non-collinear points : Three or more points are said to be non-collinear, if
they are not lying on a single straight line. Here, A, B, and C are non-collinear points. Angle : When two rays originates from the same end point, they form an
angle. The rays are called arms of an angle and common end point is called its
vertex. If the sum of the measures of two angles is 90° , then they are called complementary angles. e.g., ∠ A = 60°, ∠ B = 30° Then, ∠ A + ∠ B = 60° + 30° = 90°
30°
60°
Hence, ∠ A + ∠ B are complementary angles. If sum of the measures of two angles is 180° , then they are called supplementary angles. e.g., ∠ A = 30°, ∠ B = 150°
30° = A�
60° = B�
Here, both the angles (∠ A + ∠ B = 30° + 150° = 180°) sum is equal to 180°, hence, they are supplementary angles.
Two angles are said to be adjacent angles, if they have common vertex, a common arm and their non-common arms being on either sides of the common arm.
e.g., Here, x and y are adjacent angles. When two lines intersect, then vertically opposite angles so formed are
always equal. If the non-common sides of a pair of adjacent angles are in a straight line, then the angles form a linear pair. Intersecting lines : If two lines have a point in common, then lines are intersecting lines and common point is
called point of intersection.DA
O
BC
e.g., In the adjoining figure, AB and CD are intersecting lines and O is the point of intersection. Parallel lines : If two lines never cut each other and have no common point when produced indefinitely in either
direction, they are called parallel lines.A B
C D
Transversal Line : A line that intersects, two or more lines at distinct points is called a transversal.
(q is a transversal of l and m) Corresponding Angles : It two parallel lines are cut by a transversal, each pair of corresponding angles are equal
in measure. In the adjacent figure, the pair of corresponding angles are ∠ 1 = ∠ 5, ∠ 2 = ∠ 6, ∠ 3 = ∠ 7, and ∠ 4 = ∠ 8.
p
m
l
1 2
3 4
5 6
7 8
Alternate Angles : The alternate angles are divided into two parts : (a) Alternate interior angle : The pair of angles on opposite sides of the transversal but inside the two lines are
called alternate interior angles. (b) Alternate exterior angle : The pair of angles on the opposite sides of the transversal but outside the two lines
are called alternate exterior angles. When a transversal intersects, a set of parallel lines, then (a) the alternate angles are equal to each other. (b) the corresponding angles are equal to each other. (c) the sum of the two interior angles on the same side of the transversal is 180°. If a transversal intersects two given lines such that (i) the alternate angles are equal to each other, or (ii) the
corresponding angles are equal to each other, or (iii) the sum of two interior angles on the same side of transversal is 180° , then the given lines are parallel.
The angle formed by two lines or line segments can be a acute angle, when the measure of the angle is less than 90°.
The angle formed by two lines of line segments can be a right angle, when the measure of the angle is 90° .
The angle formed by two lines or line segment can be an obtuse angle, when the measure of the angles is greater than 90°.
The angle formed by two lines or line segment can be a reflex angle, when the measure of the angle is greater than 180°.
GREENBOARD ?How it is done on
Q.
Out of a pair of complementary angles, one is two-third of the other. Find the angles.
Sol. Step-I : Let one angle be x. So, other angle = 90° - x
Step-II : Thus, 2
3 × x = 90° - x
or 2x = 270° - 3x or 2x + 3x = 270° Step-III : or 5x = 270°
or x = 270
5 = 54°
Step-IV : So, one angle = 54° and the other angle = 90° - 54°= 36°.
Let’s Revise A triangle is a simple closed figure made of three line segments. It has three vertices, three sides and three angles. Let A, B and C be three non-collinear points. Then, the figure formed by the three line segments AB, BC and CA
is called a triangle with vertices A, B, C and it is denoted by ∆ABC.A
B C
A ∆ABC has : (a) three sides, namely AB, BC and CA. (b) three angles, namely ∠BAC, ∠ABC, ∠BCA and these are also denoted by ∠A, ∠B and ∠C respectively. Naming triangles by considering the length of their sides : (a) Equilateral triangle : A triangle having all sides equal, is called an equilateral triangle. (b) Isosceles triangle : A triangle having two sides equal, is called an isosceles triangle. (c) Scalene triangle : A triangle having all sides of different lengths is called a scalene triangle.
Naming triangles by considering their angles : (a) Acute triangle : A triangle each of whose angle measures less than 90° is called an acute triangle. (b) Right angled triangle : A triangle one of whose angle measures 90° is called a right angled triangle. (c) Obtuse triangle : A triangle one of whose angle measures more than 90° is called an obtuse triangle.
Acute triangle
(i)
Right angled triangle
(ii)
Obtuse triangle
(iii)
55º
65º 60º 90º 120º
Some results on triangle : (a) Each angle of an equilateral triangle measures 60°. (b) The angles opposite to equal sides of an isosceles triangle are equal. (c) A scalene triangle has no equal angles. (d) The non-equal side of an isosceles triangle is called its base, the base angles of an isosceles triangle have equal
measure. (e) A triangle has 3 medians and 3 altitudes. Median of a triangle : A median connects a vertex of a triangle to the mid-point of the opposite side. e.g.,
(a) In ∆ABC, the line segment AD joining the mid-point of BC to its opposite vertex is a median of the triangle. The line segment BE joining the mid-point of AC to its opposite vertex is also a median of the triangle. Altitudes of a triangle : The perpendicular line segment from a vertex of a triangle to its opposite side is called
and altitude of the triangle.A
B CD
A
B C
Emedian
altitude
Oswaal CBSE Chapterwise Quick Review, MATHEMATICS, Class-VII [ 11 The sum of the lengths of any two sides of a triangle is always greater than the length of the third side. The difference of the lengths of any two sides of a triangle is always smaller 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 or arms. In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares on its legs.
GREENBOARD ?How it is done on
Q. In ∆ABC, DE || BC, find the values of x, y and zA
D Ex y
z
30º 40º
B C
Sol.: Step I : DE || BC (given) ∠D = ∠B are [corresponding angles.] x = 30° Step II : ∠E = ∠C (corresponding angles) y = 40° Step III : In ∆ADE, x + y + z = 180° (angle sum properly) ⇒ 30° + 40° + z = 180° ⇒ z = 180° – 70° = 110°
Chapter - 7 : Congruence of Triangles
Let’s Revise Line segment : The shortest distance between two points is called a line segment. Measure : The opening between the arms of an angle is called its measure. Congruent figures : 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. Types of congruent figures : l Two line segments are congruent, if they have the same length.
A B C D
i.e., segment AB ≅ segment CD, if AB = CD. l Two angles are congruent, if they have the same measure.
A
O
B
C
DE
i.e., ∠AOB ≅ ∠CDE, if measure of ∠AOB = measure of ∠CDE. l Two squares are congruent, if they have the same side length.
A B
CD
E F
GH
l Two rectangles are congruent, if they have the same length and breadth.
A B
CD
E F
H G
i.e., rectangle ABCD ≅ rectangle EFGH, If AB = EF and AD = EH.
12 ] Oswaal CBSE Chapterwise Quick Review, MATHEMATICS, Class-VII l Two circles are congruent, if they have the same radius.
r r
C1 C2
i.e., circle C1 ≅ circle C2, if radius of C1 = radius of C2.
Criteria for congruence of triangles :
l S.S.S. Congruence Criterion :
If three sides of one triangle are respectively equal to the three sides of another triangle under a given correspondence, then the triangles are congruent.
A
B
C
Q
R P
where, AB = QR, AC = PR, BC = PQ. l S.A.S. Congruence Criterion :
If under a correspondence two sides and the angle included between these sides of a triangle are respectively equal to two sides and the angle included them of another triangle, then the triangles are congruent.
A
B
C P
Q
R
where, AB = RQ, CB = PQ, ∠ CBA = ∠ PQR l A.S.A. Congruence Criterion :
When two angles and the included side of a triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent under a correspondence.
A
B C
P
Q R
where, ∠ ABC = ∠ PQR, ∠ ACB = ∠ PRQ, BC = QR. l R.H.S. Congruence Criterion :
When the hypotenuse and one side of a right-angled triangle are respectively equal to the hypotenuse and one side of another right angled triangle under a correspondence, then the triangles are congruent.
Ex. ABC is an isosceles triangle with AB = AC, take a trace-copy of ∆ABC and also name it as ∆ACB
(a) State the three pairs of equal parts is ∆ABC and ∆ACB
(b) Is ∆ABC = ∆ACB ?
(c) Os ∠B = ∠C ?
Sol.: Step. I : We have AB = AC
Step. II : On trace copy of ∆ABC, we get ∆ACBA
B C
A
BC
Step. III : In ∆ABC and ∆ACB, we have
AB = AC, BC = CB, AC = AB
which are the three pairs of equal parts in ∆ABC and ∆ACB.
Step. IV : In ∆ABC and ∆ACB, we have
AB = AC, BC = CB, AC = AB
[since, all the three sides of ∆ABC
are equal to three sides of ∆ACB.]
∴ ∆ABC ≅ ∆ACB [by SSS congruence rule]
and A ↔ A, B ↔ C, C ↔ B
Step. V : Yes
∆ABC ≅ ∆ACB
∴ ∠B = ∠C [since, corresponding part of
congruent triangles are equal]
Chapter - 8 : Comparing Quantities
Let’s Revise If two fractions of a quantity are equal, then they are called in the ratio 1 : 1. Various quantities (same type) can be compared by using their ratios. If the two ratios are equal, the four quantities are called in proportion. To convert percent into decimals, drop the sign of percent and shift the decimal point two places to the left. To convert a fraction into percent, multiply the fraction by 100 and write % sign. To convert a decimal into percent, shift the decimal point two places to the right side and write % sign. Profit = S.P. – C.P., (S.P. > C.P.), C.P. ® Cost price and S.P. ® Selling price Loss = C.P. – S.P. (C.P. > S.P.)
Simple interest (S.I.) = P×R×T
100, P — Principal, R — Rate of interest and T — time
Amount = Principal + Interest.
1
1100
0 01% ·= =
where, 1% is percentage conversion.
1100
is a fraction conversion.
and, 0·01 is decimal conversion.
Profit %=
ProfitC.P.
×100%
Loss%=
LossC.P.
×100%
To compare quantities, there are multiple methods. Some of the methods that we are going to discuss in this chapter are – ratio and proportion, percentage, profit, loss and simple interest.
l Ratio : The ratio of two quantities of the same kind and in the same unit is the fraction that one quantity is of the other.
14 ] Oswaal CBSE Chapterwise Quick Review, MATHEMATICS, Class-VII The ratio a is to b as is the fraction a / b, . and it is written as a : b. l Terms of a ratio : In the ratio a : b, we call a as the first term or antecedent and b the second term or consequent. l Equivalent ratios : To compare different ratios, firstly convert fractions into like fractions. If like fractions are
equal, then the given ratios are said to be equivalent. l To compare two quantities, the units must be the same. A ratio remains unchanged, if both of its terms are multiplied or divided by the same non-zero quantity. e.g., Let m ≠ 0. Then, clearly we have
(a)
ab
=
mamb
i.e., (a : b) = (ma : mb)
(b)
ab
=
ambm
i.e., (a : b) =
am
bm
:
l Ratio of simplest form : The ratio (a : b) is said to be in simplest form, if HCF of a and b is 1. i.e. no common factor between a and b other than 1. It is also called the lowest term.
Note : The ratio of two numbers is usually expressed in simplest form. l Proportion : Four numbers a, b, c, d are said to be in proportion, if a : b = c : d and we write a : b : : c : d. (a) If a : b : : c : d, then (a) a, b, c, d are respectively known as first, second, third and fourth term. (b) a and d are called extremes while b and c are called means of the proportion. (c) (a × d) = (b × c) i.e. Product of extremes = Product of means (d) d is called the fourth proportional of a, b and c. (b) If a : b : : b : c, then (a) a, b, c are said to be in continued proportion. (b) c is called third proportional to a, b and fourth proportional to a, b, b.
(c)
ab
bc
= ⇔ b2 = ac ⇔ b = ac
where, b is called the mean proportional or geometric mean between a and c. Unitary method : A method in which the value of unit quantity is first obtained to find the value of any required
quantity is called unitary method. In solving problems based on unitary method, we come across two types of variations. (a) Direct Variation : Two quantities a and b are said to vary directly, if the ratio a / b remains constant e.g., (a) The cost of articles varies directly as the number of articles. (More articles, more cost), (Less articles, less cost). (b) The work done varies directly as the number of men doing work. (More men doing work, more work), (Less men doing work, less work) (b) Inverse variation : Two quantities a and b are said to vary inversely if their product a × b remains constant.
GREENBOARD ?How it is done on
Q. Rohan purchases a table lamp for ` 800. He marked up the price by ` 320 and sell. Find his gain percentage,
Note : (i) Zero is an integer, which is neither positive nor negative.
(ii) A positive integer is the same as a natural number.
Fractions : The numbers of the form ab
, where a and b (≠ 0) are whole numbers, are called fraction. e.g., 23
38
, ,
115
10223
, ,... etc.
Note : In ab
, the integer a in the numerator and the integer b (≠ 0) is the denominator.
Rational Numbers : The number of the form
pq , where p and q are integers and q ≠ 0, are called rational numbers.
e.g., 45
57
327
, ,−
,... etc., are all rational numbers.
Important points about rational numbers :
(i) Zero is a rational number. We can write 0 = 01
, which is in the form of numerator and denominator.
(ii) Every natural number is a rational number but every rational number is not a natural number.
(iii) Every integer is a rational number but a rational number need not be an integer.
(iv) Every fraction is a rational number.
Equivalent rational number : A rational number can be written with different numerators and denominators. By multiplying the numerator and denominator of a rational number by the same non-zero integer, we obtain another rational number equivalent to the given rational number.
e.g., −37
=
− ××
=−3 2
7 26
14
Same as multiplication, the division of the numerator and denominator by the same non-zero integer, also gives equivalent rational numbers.
e.g., −1664
=
− ÷÷
=−16 16
64 161
4
Note : −
−3
737
or is written as – 37
.
Positive rational number : A rational number is said to be positive, if its numerator and denominator are either both positive or both negative.
e.g., Each of the numbers 57
138
179
7240
3663
, , , ,−−
−−
is a positive rational number.
Negative rational number : A rational number is said to be negative, if its numerator and denominator are such that one of them is a positive integer and the other is a negative integer.
e.g., Each of the number −
−−
−3
535
187
187
, , , is a negative rational number.
Note : 0 (zero) is neither a positive nor a negative rational number.
Let’s Revise We can draw a parallel line to a given line. Through the point outside the line with the help of equal alternate
angles and equal corresponding angles made by a transversal line. Construction of a triangle is possible only when the sum of the lengths of any two sides of the triangle is greater
than the third side. Construction of triangle is possible by the following criterion : (a) Side-Side-Side (S.S.S.) criterion. (b) Side-Angle-Side (S.A.S.) criterion. (c) Angle-Side-Angle (A.S.A.) criterion. (d) Right-Angled-Hypotenuse (R.H.S.) criterion.
GREENBOARD ?How it is done on
Q. 1. Draw a line, say AB, take a point C outside it. Through C, draw a line parallel to AB using ruler and compasses only.
Sol. To draw a line parallel to AB using ruler and compasses, we use the following steps.
Steps of construction :
Step-I : Draw a line segment AB and then take any point P on it.
A P B
Step-II : Take a point C outside AB and join PC.
A P B
C
Step-III : With P as centre, draw an arc cutting AB and PC at X and Y respectively.
A P B
Q
C
X
Y
Step-V : Place the point P of the compasses at X and adjust the opening, so that the pencil tip is at Y with this opening and with Q as centre, draw an arc cutting the arc drawn in Step-IV at R.
A P B
RC
Q
Y
X
Step-VI : Join CR and produce it on both sides to obtain the required line l.
Step-IV : With C as centre and the same radius as in Step III, draw an arc on the opposite side of PX to cut PC at Q.
A P B
R C
Q
Y
X
l
Thus, l || AB.
Chapter - 11 : Perimeter and Area
Let’s Revise Perimeter is the distance around a closed figure whereas area is the part of plane occupied by the closed figure. Conversion of units
Length units Area units
1 cm = 10 mm 1 cm2 = (10 × 10) mm2 = 100 mm2
1 dm = 10 cm 1 dm2 = (10 × 10) cm2 = 100 cm2
1 m = 10 dm 1 m2 = (10 × 10) dm2 = 100 dm2
1 m = 100 cm 1 m2 = (100 × 100) cm2 = 10000 cm2
1 hm = 100 m 1 hm2 = (100 × 100) m2 = 10000 m2
1 km = 1000 m 1 km2 = (1000 × 1000) m2 = 106 m2
2 km = 2000 m 1 hec = 10000 m2
Area and perimeter of a rectangle : Consider a rectangle with length = l units and breadth = b units. Then, we have (a) Area of the rectangle = (l × b) sq units,
l
lA B
CD
b b where, length =
AreaBreadth
units
and breadth = Area
Length
units
(b) Diagonal = l b2 2+ units
(c) Perimeter = 2(l + b) units
Area of four walls of a room, Let there be a room with length = l units, breadth = b units height = h units. Then, we have
(a) Area of the four walls = [2(l + b) × h] sq units
(b) Diagonal of the room = l b h2 2 2+ +
Area and perimeter of a square Let there be a square each of whose sides measures a units. Then, we have
(a) Area of the square = a2 sq. units a
a
a a
A B
CD
(b) Side of the square = Area( ) units
(c) Diagonal of the square = 2a( ) units
(d) Area of square = 12
2×( )
Diagonal
(e) Perimeter of the square = (4a) units
18 ] Oswaal CBSE Chapterwise Quick Review, MATHEMATICS, Class-VII Area of the triangles : A closed plane figure made by three line
segments is called a triangle.
(a) Consider a triangle in which the base and the corresponding height altitude (or height) are given.
Then, Area of the triangle = 12
× ×
Base Height Altitude( ) sq. units
units
A
B CD
Base
Altitude
(b) In a right-angled triangle, the sides other than the hypotenuse (side opposite to right angle) are called its legs. Then, Area of a right triangle
= 12
×
Product of legs sq. units
(c) Let each side of an equilateral triangle be a units. Then,
A
B C
Area of the equilateral triangle = 3
42a
sq. units
A
B C
a a
a Note : All the congruent triangles are equal in area but the triangles
equal in area need not be congruent.
Area of a parallelogram : A quadrilateral in which opposite sides in pairs are parallel, is called a parallelogram
Area of a parallelogram = (AB × DE) sq units
A B
CD
E
= (Base × Height) sq units
Circle : A round plane figure whose boundary (the circumference) consists of points equidistant from a fixed point (the centre).
Chord : A line segment joining any two points on a circle is called a chord of the circle.
Diameter : A chord passing through the centre of a circle is known as its diameter.
Circumference : The perimeter of a circle is called its circumference.
∴ Circumference of a circle = 2pr.
The ratio of circumference and diameter of a circle is a constant and is denoted by p (pi).
Approximate value of p is taken 227
or 3.14.
r
Circumference of a circle of radius r is 2pr.
GREENBOARD ?How it is done on
Q. A nursery school play ground is 160 m long and 80 m wide. In it 80 m × 80 m is kept for swings and in the remaining portion, there is 1.5 m wide path par-allel to its width and parallel to its remaining length as shown in Fig. The remaining area is covered by grass. Find the area covered by grass.
1.5 m
Swings 80 m
80 m80 m
1.5 m
Oswaal CBSE Chapterwise Quick Review, MATHEMATICS, Class-VII [ 19Sol.: Step–1 : Area of school playground is 160 m × 80 m = 12800 m2 Area kept for swings = 80 m × 80 m = 6400 m2
= 80 m × 1.5 m = 120 m2
Step–II : Area of path parallel to the remaining length of playground
= 80 m × 1.5 m = 120 m2
Step–III : Area common to both paths = 1.5 m × 1.5 m = 2.25 m2
[since it is taken twice for measurement it is to be subtracted from the area of paths]
Step–IV : Total area covered by both the paths
= (120 + 120 – 2.25) m2
= 237.75 m2
Step–V : Area covered by grass = Area of school playground – (Area kept for swings + Area covered by paths)
= 12800 m2 – [6400 + 237.75] m2
= (12800 – 6637.75) m2
= 6162.25 m2
Chapter - 12 : Algebraic Expression
Let’s Revise Algebraic expression is formed from variables and constants using different operations. Expressions are made up of terms. A term is the product of factors. Factors may be numerical as well as algebraic (literal). Coefficient is the numerical factor in a term. Sometimes, any factor in a term is called the coefficient of the
remaining part of the term. The terms having the same algebraic factors are called like terms. The terms having different algebraic factors are called unlike terms. Expression with one term is called a ‘Monomial’. Expression with two unlike terms is called a ‘Binomial’ Expression with three unlike terms is called a ‘Trinomial’. In general, an expression with one or more than one term (with non-negative integral exponents of the variables)
is called a ‘Polynomial’. The sum (or difference) of two like terms is a like term with coefficient equal to the sum (or difference) of
coefficients of the two like terms. When we add (or subtract) two algebraic expressions, the like terms are added (or subtracted) and the unlike
terms are written as they are. To find the value of an expression, we substitute the values of the variables in the expression and then simplify. Rules and formulae in mathematics are written in a concise and general form using algebraic expressions.
GREENBOARD ?How it is done on
Q. Simplify the expression by combining the like terms :
7x3 – 3x2y + xy2 + x2y – y3
Sol. Step-I : Rearranging the terms in the given expression, we get
Here 10, 3 and 2 are the bases, whereas 5, 4 and 7 are their respective exponents. We also say, 1,00,000 is the 5th power of 10; 81 is the 4th power of 3 etc.
Numbers in exponential form obey certain laws, which are : For any non-zero integers a and b and whole numbers m and n : (a) am × an = am + n
(b) am ÷ an = am – n
(c) (am)n = amn
(d) am × bm = (ab)m
(e) am ÷ bm = ab
m
(f) a0 = 1 (g) (– 1)even number = 1 (h) (– 1)odd number = – 1
GREENBOARD ?How it is done on
Q. Find x so that 53
53
5 11
×
=
53
8
x
Sol.: Given, 53
53
5 11
×
=
53
8
x
Step-I : So, 53
53
5
5
11
11× = 53
8
x
Usingab
ab
m m
m
=
or 5 53 3
5 11
5 11××
= 53
8
x
Step-II : 53
16
16( )( )
= 53
8
x
{Using am × an = (a)m + n}
Step-III : 53
16
=
53
8
x
Step-IV : 16 = 8x
Thus, 8x = 16
Step-V : Therefore, x = 2
Chapter - 14 : Symmetry
Lets’ Revise Symmetry : A figure has line of symmetry if there is a line about which the figure may be folded so that the two
parts of figure will coincide. The line of symmetry is also called axis of symmetry. An isosceles triangle is symmetrical about the bisector of the angle included between the equal sides. e.g., In the given figure, ∆ABC is given in which AB = AC and AD is the bisector of ∠BAC. Then, AD is the line of
symmetry of ∆ABC.
B CD
A
An equilateral triangle is symmetrical about each one of the bisectors of its interior angles. e.g., In the given figure, ∆ABC is an equilateral triangle and lines AD, BE and CF are bisectors. Then, AD, BE and CF are lines of symmetry.
A square is symmetrical about the diagonals and the lines joining the mid-points of its opposite side, e.g. In the given figure, ABCD is a square in which AC and BD are diagonals and lines FE and GH are bisector lines. Then, AC, BD, FE and GH are lines of symmetry.
C
H
G
FD
BEA
A rectangle has two lines of symmetry, each one of which is the line joining the mid-points of its opposite sides. e.g., In the given figure, rectangle ABCD is symmetrical about PQ and RS.
D
R
A
QC
S
BP
A rhombus is symmetrical about each one of its diagonals. e.g., In the given figure, ABCD is a rhombus AC and BD are diagonals. Then, AC and BD are lines of symmetry.
D
A B
C
A kite is symmetrical about one of its diagonal. e.g., In the given figure, ABCD is a kite in which AB = AD and BC = DC and AC is a diagonal. Then, diagonal AC is a line of symmetry.
A
B D
C
A trapezium is symmetrical about the line joining the mid-points of the parallel line, e.g., In the given figure, ABCD is a trapezium in which AB || DC and AD = BC. E and F be the mid-points of AB and DC respectively. Then, trapezium ABCD is symmetrical about EF.
A B
CD
E
F
UB
A circle is symmetrical about each one of its diameters. e.g., In the given figure, a circle has many diameters. Then, all diameters are lines of symmetry.
22 ] Oswaal CBSE Chapterwise Quick Review, MATHEMATICS, Class-VII A semi-circle is symmetrical about its perpendicular bisector of diameter, e.g., In the given figure, semi-circle ABC
in which AB is a diameter and PCQ is a perpendicular bisector. Then, PQ is line of symmetry.
P
C
AQ B
Each of the following capital letters of the English alphabet is symmetrical about the dotted line or lines as shown :
A B C D E H I K
M O T U V W X Y
Line symmetry, rotational symmetry and order of rotational symmetry of various figures :
Figure names Linesymmetry
Number of symmetry
Rotationalsymmetry
Centre ofrotation
Order of rotationalsymmetry
Square Yes 4 Yes Point of intersection of diagonals
4
Rectangle Yes 2 Yes Intersection of diagonals
2
Equilateral Triangle
Yes 3 Yes Centroid 3
Regular Hexagon
Yes 6 Yes Centre of the hexagon
6
Circle Yes Infinite Yes Centre Infinite
Parallelogram No 0 Yes Intersection of diagonals
2
Rhombus Yes 2 Yes Intersection of diagonals
2
GREENBOARD ?How it is done on
Q. Which of the following figures have rotational symmetry of order more than 1 ?
(a) (b) (c)
Sol. (a) : Figure (a) has rotational symmetry of order 4 about the marked point through an angle of 90°.
(b) : Figure (b) has rotational symmetry of order 3 about the marked point through an angle of 120°.
(c) : Figure (c) has no rotational symmetry about the marked point.
(d) : Figure (d) has rotational symmetry of order 2 about the marked point through an angle of 180°.
(e) : Figure (e) has rotational symmetry of order 3 about the marked point through an angle of 120°.
(f) : Figure (f) has rotational symmetry of order 4 about the marked point through an angle of 90°.
Hence, figures (a), (b), (d), (e) and (f) have rotational symmetry of order more than 1.
Chapter - 15 : Visualising Solid Shapes
Lets’ Revise Figures drawn on paper are called plane figure. These figures are circle , triangle , square , rectangle ,
etc. Solid figures like sphere, cone, cylinder, cube, cuboid, etc. occupy space. Plane figures are 2-dimensional, while solids are 3-dimensional. The flat surfaces that form the skin of solids are called faces. The line segments that form the skeleton of solids are called edges. The points where these edges meet are called vertices. 3-dimensional figures can be represented as 2-dimensional figures by drawing their oblique or isometric sketches. The corner of solid figures are called vertices. Line segments joining vertices are called edges and their surfaces
are called faces. Different sections of a solid are viewed either by seeing it or observing its 2-D shadow. It can also viewed from
front, top or side. A net is a skeleton outline of a solid that can be folded to make it. A solid can have more than one net. Different sections of a solid can be viewed in many ways : (a) By cutting or slicing, the shape, which would result in the cross-section of the solid. (b) By observing a 2-D shadow of a 3-D shape. (c) By looking at the shape from different position the front-view, the side-view and the top-view.
GREENBOARD ?How it is done on
Q. What cross-sections do you get when you give a : (a) vertical cut (b) horizontal to the following solids : (a) A duster (b) A football (c) A dice (d) A brick (e) A circular pipe (f) An ice-cream cone (g) a round apple.Sol.: (a) A duster : (i) Vertical cut : cross-section may be almost
square. (ii) Horizontal cut : cross-section will be almost
rectangle. (b) A football : (i) Vertical cut : cross-section will be almost a
circle. (f) An ice-cream cone : (i) Vertical cut : cross-section will be almost a
semi-triangle. (ii) Horizontal cut : cross-section will be almost a
circle.
(ii) Horizontal cut : cross-section will be almost a circle.
(c) A dice : (i) Vertical cut : cross will be almost a square. (ii) Horizontal cut : cross-section will be almost a
square. (d) A brick : (i) Vertical cut : cross-section will be almost a
square. (ii) Horizontal cut : cross-section will be almost a
rectangle. (e) A circular pipe : (i) Vertical cut : cross-section will be almost a
semi-circle. (ii) Horizontal cut : cross-section will be almost a
rectangle. (g) A round apple : (i) Vertical cut : cross-section will be almost a
circle. (ii) Horizontal cut : cross-section will be almost a
