The word ‘geometry’ originally come from the Greek word ‘geo’ meaning ‘earth’ and ‘metron’ meaning ‘measurement’. The word geometry means ‘measurement of earth’ or the science of properties and relation of figures.
Point : A point determine allocation ‘.’
Line : A line is a straight path that extends indefinitely in both direction.
Intersecting Lines : If two lines have one common point, they are called intersecting line
Parallel Lines :
Triangle :
6.Square :
Rectangle :
Polygons :
Circle :
Example : Consider the given figure and answer the following questions : (i) Is it a curve ? (ii) Is it a closed figure?
Solution : (i) It is not a curve . It is a polygon entirely made up of the line segment . (ii) Yes, It is closed figure.
When we fold a paper in such a way that the picture is divided into two equal halves then the line which divides the picture into two halves is called a Line of Symmetry.
Here the line divides the star into two halves so it is the line of symmetry. It is also called the Mirror Line because if we place the mirror on that line then one side of the picture will fall exactly on the other side of the picture.
Non-symmetrical Figure
This figure is not symmetrical as if we fold the image from the dotted line then it does not divide it into two equal halves.
Making Symmetric Figures: Ink-blot Devils
To make an ink-blot pattern-
Take a piece of paper and fold it in half.
Put some drops of ink on one side of the paper.
Then press the halves together.
It will make a symmetric pattern with the fold as the line of symmetry.
Inked-string pattern
To make an inked string pattern-
Take a piece of paper and fold it in half.
Dip a string in different colours and arrange it on the one side of the paper.
Press the two halves together and pull the string.
It will make a symmetric inked string pattern with the fold as the line of symmetry.
Two Lines of Symmetry
Some figures have two lines of symmetry.
1. A Rectangle
Take a rectangular sheet and fold it horizontally in two equal halves and then again fold it vertically in two equal halves. After opening it, we get two lines of symmetry of the rectangular sheet.
2. More Figures with two Line of Symmetry
If we take a rectangular piece of paper and double fold it to make two lines of symmetry and cut it in some new shape then after opening it we will get a new image that too with the two lines of symmetry.
Construction of figure with two Lines of Symmetry
1. To draw a figure with two lines of symmetry, take one figure.
2. Let L and M be the two lines of symmetry.
3. Draw the figure in such a way that L is the line of symmetry,
4. Now complete the figure by drawing the remaining part so that M will also become the line of symmetry.
Hence this is the final figure with two lines of symmetry.
Multiple Lines of Symmetry
Take a square sheet of paper and fold it in two halves vertically and again horizontally .open it and fold it in two equal halves diagonally then again open it and fold it along another diagonal.
When you will open the paper you will see four imaginary lines and these lines are the lines of symmetry.
Some more images with more than two lines of symmetry
Equilateral triangle will have three lines of symmetry.
Square will have four lines of symmetry.
Regular pentagon will have five lines of symmetry.
Regular hexagon will have six lines of symmetry.
Some Real-life Examples of Symmetry
In Taj Mahal and the butterfly there is one line of symmetry and there are so many other things also in our daily life which are having one or more line of symmetry.
Reflection and Symmetry
The line of symmetry is also called Mirror Line because the mirror image of an object is symmetrical to the image. When we see an object in the mirror then there is no change in the length and angles of the object except one thing i.e. the image is opposite to the original image.
Some Examples of Reflection Symmetry
1. Paper Decoration
We can use a rectangular sheet to fold and create some intricate patterns by cutting paper.
2. Kaleidoscope
In Kaleidoscope, mirrors are used to create pictures having various lines of symmetry. Two mirrors strips forming a V-shape are used. The angle between the mirrors determines the number of lines of symmetry.
Example
Which alphabet will remain same after reflection symmetry? Check for R, C, N, A and T.
Solution
In the alphabet reflection symmetry, the alphabets look opposite in the mirror i.e. the alphabet written from right to left will appear as written from left to right.
Hence C, N and R will not look the same after reflection.
Hence A and T will look same after reflection symmetry.
It is a simple closed curve and is not considered as a polygon.
Parts of Circles
1. Radius – It is a straight line connecting the centre of the circle to the boundary of the same. Radii is the plural of ‘radius’.
2. Diameter –It is a straight line from one side of the circle to the other side passing through the centre.
3. Circumference – It refers to the boundary of the circle.
4. Chord – Any line that connects two points on the boundary of the circle is called Chord. Diameter is the longest chord.
5. Arc – It is the portion of the boundary of the circle.
6. Interior of the Circle – Area inside the boundary of the circle is called the Interior of the Circle.
7. The Exterior of the Circle – Area outside the boundary of the circle is called the Exterior of the Circle.
8. Sector– It is the region in the interior of a circle enclosed by an arc on one side and a pair of radii on the other two sides.
9. Segment – It is the region in the interior of the circle enclosed by an arc and a chord.
Semi-circle
A diameter divides the circle into two semi-circles. Hence the semicircle is the half of the circle, which has the diameter as the part of the boundary of the semicircle.
Construction of a circle if the radius is known
Draw a circle of radius 5 cm.
To draw a circle, we need a compass and a ruler to measure the length.
Step 1: Open the compass and measure the length of 5 cm using a ruler.
Step 2: Mark a point O, which we will use as the centre of the circle.
Step 3: Put the pointer on the point O.
Step 4: Turn the compass to make a complete circle. Remember to do it in one instance.
Prime Factorisation is the process of finding all the prime factors of a number.
There are two methods to find the prime factors of a number-
1. Prime factorisation using a factor tree
We can find the prime factors of 70 in two ways.
The prime factors of 70 are 2, 5 and 7 in both the cases.
2. Repeated Division Method
Find the prime factorisation of 64 and 80.
The prime factorisation of 64 is 2 × 2 × 2 × 2 × 2 × 2.
The prime factorisation of 80 is 2 × 2 × 2 × 2 × 5.
Highest Common Factor (HCF)
The highest common factor (HCF) of two or more given numbers is the greatest of their common factors.
Its other name is (GCD) Greatest Common Divisor.
Method to find HCF
To find the HCF of given numbers, we have to find the prime factorisation of each number and then find the HCF.
Example
Find the HCF of 60 and 72.
Solution:
First, we have to find the prime factorisation of 60 and 72.
Then encircle the common factors.
HCF of 60 and 72 is 2 × 2 × 3 = 12.
Lowest Common Multiple (LCM)
The lowest common multiple of two or more given number is the smallest of their common multiples.
Methods to find LCM
1. Prime Factorisation Method
To find the LCM we have to find the prime factorisation of all the given numbers and then multiply all the prime factors which have occurred a maximum number of times.
Example
Find the LCM of 60 and 72.
Solution:
First, we have to find the prime factorisation of 60 and 72.
Then encircle the common factors.
To find the LCM, we will count the common factors one time and multiply them with the other remaining factors.
LCM of 60 and 72 is 2 × 2 × 2 × 3 × 3 × 5 = 360
2. Repeated Division Method
If we have to find the LCM of so many numbers then we use this method.
Example
Find the LCM of 105, 216 and 314.
Solution:
Use the repeated division method on all the numbers together and divide until we get 1 in the last row.
LCM of 105,216 and 314 is 2 × 2 × 2 × 3 × 3 × 3 × 5 × 7 × 157 = 1186920
Real life problems related to HCF and LCM
Example: 1
There are two containers having 240 litres and 1024 litres of petrol respectively. Calculate the maximum capacity of a container which can measure the petrol of both the containers when used an exact number of times.
Solution:
As we have to find the capacity of the container which is the exact divisor of the capacities of both the containers, i. e. maximum capacity, so we need to calculate the HCF.
The common factors of 240 and 1024 are 2 × 2 × 2 × 2. Thus, the HCF of 240 and 1024 is 16. Therefore, the maximum capacity of the required container is 16 litres.
Example: 2
What could be the least number which when we divide by 20, 25 and 30 leaves a remainder of 6 in every case?
Solution:
As we have to find the least number so we will calculate the LCM first.
LCM of 20, 25 and 30 is 2 × 2 × 3 × 5 × 5 = 300.
Here 300 is the least number which when divided by 20, 25 and 30 then they will leave remainder 0 in each case. But we have to find the least number which leaves remainder 6 in all cases. Hence, the required number is 6 more than 300.
The closed 2-D shapes are referred to as plane figures.
Here “C” is the boundary of the above figure and the area inside the boundary is the region of this figure. Point D comes in the area of the given figure.
Perimeter
If we go around the figure along its boundary to form a closed figure then the distance covered is the perimeter of that figure. Hence the Perimeter refers to the length of the boundary of a closed figure.
If a figure is made up of line segments only then we can find its perimeter by adding the length of all the sides of the given figure.
Example
Find the Perimeter of the given figure.
Solution
Perimeter = Sum of all the sides
= (12 + 3 + 7 + 6 + 10 + 3 + 15 + 12) m
= 68 m
The Perimeter of a Rectangle
A rectangle is a closed figure with two pairs of equal opposite sides.
Perimeter of a rectangle = Sum of all sides
= length + breadth + length + breadth
Thus, Perimeter of a rectangle = 2 × (length + breadth)
Example: 1
The length and breadth of a rectangular swimming pool are 16 and 12 meters respectively .find the perimeter of the pool.
Solution:
Perimeter of a rectangle = 2 × (length + breadth)
Perimeter of the pool = 2 × (16 + 12)
= 2 × 28
= 56 meters
Example: 2
Find the cost of fencing a rectangular farm of length 24 meters and breadth 18 meters at 8/- per meter.
Solution:
Perimeter of a rectangle = 2 × (length + breadth)
Perimeter of the farm = 2 × (24 + 18)
= 2 × 42
= 84 meter
Cost of fencing = 84 × 8
= Rs. 672
Thus the cost of fencing the farm is Rs. 672/-.
Regular Closed Figure
Figures with equal length of sides and an equal measure of angles are known as Regular Closed Figures or Regular Polygon.
Perimeter of Regular Polygon = Number of sides × Length of one side
Perimeter of Square
Square is a regular polygon with 4 equal sides.
Perimeter of square = side + side + side + side
Thus, Perimeter of a square = 4 × length of a side
Example
Find the perimeter of a square having side length 25 cm.
Solution
Perimeter of a square = 4 × length of a side
Perimeter of square = 4 × 25
= 100 cm
Perimeter of an Equilateral Triangle
An equilateral triangle is a regular polygon with three equal sides and angles.
Perimeter of an equilateral triangle = 3 × length of a side
Example
Find the perimeter of a triangle having each side length 13 cm.
Solution
Perimeter of an equilateral triangle = 3 × length of a side
Perimeter of triangle = 3 × 13
= 39 cm
Perimeter of a Regular Pentagon
A regular pentagon is a polygon with 5 equal sides and angles.
Perimeter of a regular pentagon = 5 × length of one side
Example
Find the perimeter of a pentagon having side length 9 cm.
Solution
Perimeter of a regular pentagon = 5 × length of one side
Perimeter of a regular pentagon = 5 × 9
= 45 cm
Perimeter of a Regular Hexagon
A regular hexagon is a polygon with 6 equal sides and angles.
Perimeter of a regular hexagon = 6 × Length of one side
Example
Find the perimeter of a hexagon having side length 15cm.
Solution
Perimeter of a regular hexagon = 6 × Length of one side
Perimeter of a regular hexagon = 6 × 15
= 90 cm
Perimeter of a Regular Octagon
A regular octagon is a polygon with 8 equal sides and angles.
Perimeter of a regular octagon = 8 × length of one side
Example
Find the perimeter of an octagon having side length 7cm.
Solution
Perimeter of a regular octagon = 8 × length of one side
Perimeter of a regular octagon = 8 × 7
= 56 cm
Area
Area refers to the surface enclosed by a closed figure.
To find the area of any irregular closed figure, we can put them on a graph paper with the square of 1 cm × 1 cm .then estimate the area of that figure by counting the area of the squares covered by the figure.
Here one square is taken as 1 sq.unit.
Example
Find the area of the given figure. (1 square = 1 m2)
Solution
The given figure is made up of line segments and is covered with some full squares and some half squares.
Full squares in figure = 32
Half squares in figure = 21
Area covered by full squares = 32 × 1 sq. unit = 32 sq. unit.
Area covered by half squares = 21 × (1/2) sq. unit. = 10.5 sq. unit.
Total area covered by figure = 32 + 10.5 = 42.5 sq. unit.
Area of a Rectangle
Area of a rectangle = (length × breadth)
Example
Find the area of a rectangle whose length and breadth are 20 cm and 12 cm respectively.
Solution
Length of the rectangle = 20 cm
Breadth of the rectangle = 12 cm
Area of the rectangle = length × breadth
= 20 cm × 12 cm
= 240 sq cm.
To find the length of a rectangle if breadth and area are given:
Example
What will be the length of the rectangle if its breadth is 6 m and the area is 48sq.m?
Solution
Length = 48/6
= 8 m
To find the breadth of the rectangle if length and area are given:
Example
What will be the breadth of the rectangle if its length is 8 m and the area is 81 sq.m?
Solution
Breadth = 81/8
= 9 m
Area of a Square
Area of a square is the region covered by the boundary of a square.
If we compare two quantities using division then it is called ratio. It compares quantities in terms of ‘How many times’. The symbol to represent ratio is “:”.
It reads as “4 is to 3”
It can also be written as 4/3.
Example
If there are 35 boys and 25 girls in a class, then what is the ratio of
Number of boys to total students
Number of girls to total students.
Solution
In the ratio, we want the total number of students.
Total number of students = Number of boys + Number of girls
35 + 25 = 60
Ratio of number of boys to total number of students
The ratio of the number of girls to the total number of students
The unit must be same to compare two quantities
If we have to compare two quantities with different units then we need to convert them in the same unit .then only they can be compared using ratio.
Example
What is the ratio of the height of Raman and Radha if the height of Raman is 175 cm and Radha is 1.35 m?
Solution
The unit of the height of Raman and Radha is not same so convert them in the same unit.
Height of Radha is 1.35 m = 1.35 × 100 cm = 135 cm
The ratio of the height of Raman and Radha
Equivalent Ratios
If we multiply or divide both the numerator and denominator by the same number then we get the equivalent ratio. There could be so many equivalent ratios of the same ratio.
In the case of equivalent ratios only their value changes but they represent the same portion of the quantity.
Example
Find two equivalent ratios of 2/4.
Solution
To get the equivalent ratio we multiply both the numerator and denominator with 2.
To get another equivalent ratio we divide both the numerator and denominator with 2.
From the above figure, we can see that in all the equivalent ratios only the number of equal parts is changing but all the ratios are representing the half part of the circle only.
The Lowest form of the Ratio
If there is no common factor of numerator and denominator except one then it is the lowest form of the ratio.
Example
Find the lowest form of the ratio 25: 100.
Solution
The common factor of 25 and 100 is 25, so divide both the numerator and denominator with 25.
The method in which we first find the value of a unit quantity and then use it to find the value of any required quantity is called the unitary method. The unitary method can be used to solve problems related to distance, time speed, and calculating the cost of materials. The unitary method is used for various applications.
The unitary method consists of two type of variations:
Two quantities are said to be in direct variation if one quantity increases, then the other also increases or when one quantity decreases, the other also decreases.
Two quantities are said to be inverse variation if,
On increasing one quantity, the other quantity decreases.
On decreasing one quantity, the other quantity increases.
Let us consider some examples:
Example 1:The cost of 15 pens is Rs 360, What is the cost of 8 such pens?
Solution:
Cost of 15 pens = Rs, 360.
Cost of 1 pen = Rs. 360/15.
Cost of 8 pen = (360/15) * 8 = Rs 192.
Example 2:18 men can make 90 identical tables in one day. Find how many men will make 20 such tables in one day?
Solution:
In one day, 90 tables are made by 18 men.
In one day, 1 tables are made by 18/1 men.
In one day, 20 tables are made by (18/1) * 20 men.
Example 3: A car running with uniform speed covers a distance of 96 km in 3 hours. How much distance will the car cover in 5 hours running with the same speed?
Solution:
In 3 hours, car covers 96 km.
In 1 hours, car covers km = (96/3) = 32 km.
In 5 hours, car covers = 32 * 5 = 170 km.
Example 4: A car can travel 360 km consuming 24 litres of petrol. How much petrol will it consume while travelling through a distance of 480 km?
Solution:
The car can travel 360 km consuming 24 litres of petrol.
The car can travel 1 km consuming (24/360)km.
The car can travel 480 km consuming = (24/360) * 480 = 32 litres.
Finding and understanding patterns gives us great power. With patterns we can learn to predict the future, discover new things and better understand the world around us.
And playing with patterns is fun.
Arithmetic Sequences
An Arithmetic Sequence is made by adding the same value each time.
Example
1, 4, 7, 10, 13, 16, 19, 22, 25, …
This sequence has a difference of 3 between each number. The pattern is continued by adding 3 to the last number each time, like this:
Geometric Sequences
A Geometric Sequence is made by multiplying by the same value each time.
1, 3, 9, 27, 81, 243, …
This sequence has a factor of 3 between each number. The pattern is continued by multiplying by 3 each time, like this:
The Fibonacci Sequence is found by adding the two numbers before it together. The 2 is found by adding the two numbers before it (1+1) The 21 is found by adding the two numbers before it (8+13) The next number in the sequence above would be 55 (21+34)
A number line is a picture of a graduated straight line. The integers are shown as marked points evenly spaced on the number line. The line includes all real numbers, continuing forever in each direction, and also numbers not marked that are between the integers. It is used to help in teaching simple addition and subtraction, which involves negative numbers. A number line is usually represented as being horizontal. According to one custom, positive numbers always lie on the right side of zero, negative numbers always lie on the left side of zero, and arrowheads on both ends of the line are meant to suggest that the line continues indefinitely in the positive and negative directions.
Now, let us understand some terms.
Integers: The collection of the numbers, that is, ? -3, -2, -1, 0, 1, 2, 3, ?., is called integers.
Absolute value: The distance of a rational number from zero on the number line is called its absolute value.
Natural numbers: Numbers by which we can count things in nature are called natural numbers.
Whole numbers: Natural number along with zero forms a collection of whole numbers.
Let us understand number line:
If you move towards the right from the zero mark on the number line, the value of thenumbers increases. If you move towards the left from the zero mark on the number line, the value of the numbers decreases.
A number line starting from 1 and marked 2, 3, 4, 5 at equal distances on the right side of 1 is called number line representing natural numbers.
A number line starting from 0 and marked 1, 2, 3, 4, 5 at equal distances on the right side of 0 is called number line representing whole numbers.
A number line with 0 marked anywhere on it with positive numbers 2, 3, 4, 5 marked on the right side of 0 at equal distances and negative numbers -1, -2, 4, 5 marked on the left side of ) is number line representing integers.
Let us consider some examples:
Example 1: Which is greater?
a) -12 and 15
b) -15 and -23
Solution:
a) 15
b) -15
Example 2: Replace * with the signs > or < in the given statements.
