A point is an exact location
The straight path between two points A and B is called a line segment AB. A line segment has two end points.
On extending a line segment AB indefinitely in one direction we get the ray AB. Ray AB has one end point, namely A.
A line segment AB extended indefinitely in both directions is called line AB.
In the given figure, the points A,B,C are collinear.
Three or more lines intersecting at the same points are called concurrent lines.
Two rays OA and OB having a common end points O form angle AOB, written as ∠AOB
The amount of turning from OA to OB is called the measure of ∠AOB written as m(∠AOB).
If a ray OA starting from its original position OA , rotates about O in anticlockwise direction and after a complete rotation comes back to its original position , then we say that it has rotated through 360. This complete rotation is divided into 360° equal parts. Then, each part is called 1 degree , written as 1°
1° = 60 minutes, written as 60'
1 minute = 60 seconds, written as 60"
If a ray stands on a line , than the sum of two adjacent angle so formed is 180° In the given figure , ray CP stands on line AB.
∴ ∠ACD + ∠BCD = 180°.
The sum of all angle around a point is 360° In the given figure five angle are formed around a point O.
∴∠AOB + ∠BOC + ∠COD + ∠DOE + ∠EOA=360°.
If two lines A Band CD intersect at a point O, then AOC , BOD and BOC , AOD are two pair of vertically opposites angle Vertically opposite angle are always equal.
∴ ∠AOC = ∠BOD and ∠AOD = ∠BOC
If two lines lie in the same plane and do not intersect when produced on either side then such lines are said to be paralleled and we write , Lm.
Let two parallel lines AB and CD be cut by a transversal EF. Then Corresponding angle are equal
(∠1 = ∠5), (∠4= ∠8), (∠2 = ∠6) , (∠3 = ∠7)
Alternate interior angles are equal.
(∠3 =∠5 ) and (∠4 =∠6)
Consective interior angles are supplementary
∠4+∠5 = 180° and ∠3 +∠6 = 180°.
A figure bounded by three straight lines is called a triangle. In the given figure , we have ∆ABC; ∆ABC having three vertices A,B,C. In has three angles, namely ∠A,∠B and ∠C. It has three sides , namely AB, AC and BC.
A figure bounded by four straight line is called a quadrilateral. The sum of all angles of a quadrilateral is 360°.
Data is the collection of numbers, facts, and figures which have some meaningful interpretation when arranged in a proper way,
For example, The representation of data regarding people below the poverty line helps in providing help to such people.
When analysis or interpretation of data is being made to draw some inference and conclusions, it is known as data interpretation.
Generally, the question asked from data interpretation are based on the topics ‘Percentage’, ‘Average’, ‘Addition/subtraction’ and ‘Ratio and Proportion’. So, command over calculations and these topics is necessary to score well in this section.
The different methods of arranging data are:
In the tabular method, data is arranged in vertical and horizontal rows. It is the easiest way of representing data but not the easiest way of interpreting data. Generally, questions based on tabular method comprises of data regarding Production/Profit/sales of different companies in a year, List of students in a class, list of defective items, Income of different persons, etc.
In the tabular method, either rows or columns are used to represent the discrete nonconnected data while other represents connected continuous variable.
For example: In the Representation of Production/Profit/sales of different companies in a year, the production of one company in different years represents connected continuous variables while the production of different companies represents the nonconnected discrete variables.
Strategy to solve tabular method DI
Generally, the question asked from Table Data Interpretation (DI) comprise of two kinds of tables: (1) Complete data tables (2) Missing data tables
While solving the missing data table, try to complete the data in the Table if it can be initially completed, as it will help you in solving questions.
To solve the question, first, note down all the variables against which you have to extract the data from Table. E.g., In a table chart comprising of incomes data of 5 persons in 5 years and for solving questions you need data of persons B and C in two years, then just write down all the needed data first and then start solving the question.
Questions from tablular data are basically asked in three forms:
Example 1: Directions: Read the following information carefully and answer the questions accordingly
The following Table represents the population of six different cities (in thousands) and the percentage of males, females, and children among them. It is also given that there is no other person(s) who lie outside the category of males, females, and children. Also, children are exclusive of male and female.
Q: What is the average number of children in city A, C, E and F?
Sol: Number of children in city A = 22% of 36000 = 7920
Number of children in city E = 42% of 86000 = 36120
Percentage of children in city C = (100 – 24 – 52) = 24%
Number of children in city C = 24% of 72000 = 17280
Percentage of children in city F = (100 – 44 – 25) = 31%
Number of children in city F = 31% of 94000 = 29140
Average number of children
Note: To solve this type of question, one can use approximation and eliminate other options.
There are different methods to represent data graphically. Some of them are:
This is the easiest method for both representation and interpretation of data, so it is a widely used method for representation of data. In this method, data is represented through horizontal/vertical rectangles known as a bar. The data representation is through the length of the bars, while the width of the bars does not have any meaning.
Data is represented in a scaled form. Scaling of data means the actual value or amount is shrunk or expanded to a scaled amount so that data fits in the bars easily.
Example 2: Directions: Study the following bar chart carefully and answer the questions given beside.
The bar graph given below represents the number of boys and the number of girls studying in five different colleges.
Q: What is the ratio between the sum of the number of boys studying in colleges A, B and C together to the sum of the number of girls studying in the college B, C and D together?
Sol: The number of boys studying in the college A, B, and C together = 120 + 220 + 340 = 680
The number of girls studying in the college B, C, and D together = 300 + 250 + 150 = 700
The required ratio = 680 : 700 = 34 : 35
In this chart, data is represented between Horizontal and vertical axis. Data is represented through a point between the axes, and then points are connected through lines to give it the form of a line chart.
If a line increment upwards between two points, then quantity on point 2 is said to be more than quantity on point 1, and if line decrement downwards between two points, then quantity is said to be decreasing.
The nonconnected discrete variables are represented through different line graphs between the same horizontal and vertical axis. For example: If we are representing the production of 3 companies in 5 years, then it will be represented by 3 line graphs.
Line Charts are generally used to show the growth of a company or a product.
Example 3: Directions: The line chart given below shows five dealers A, B, C, D, and E selling three different types of cars (In thousands) viz. Swift, Audi, and WagonR. Read the following line chart and solve the given question
Q: Number of cars sold by A and B is how much percentage more than the cars sold by C?
Sol: Cars sold by A & B = (65 + 55)
A&B=(65+55) thousand = 1,20,000
Cars sold by C = 90,000
= 33.33%
The central angle of each sector or part
Example 4: Directions: The following pie charts show the distribution of students of graduate and postgraduate levels in seven different institutes in a town.
Q. What is the difference between the central angles of nongraduate and graduate students in institute T?
Sol: Value of graduate students in institute T = 15%
Total value of graduate students = 100%
Value of nongraduate students in institute T = 18%
Total value of nongraduate students = 100%
Central angles of nongraduate students in institute T will be
Central angles of graduate students in institute T will be
Thus, the required difference = central angles of nongraduate students in institute T  Central angles of graduate students in institute T will be
In the Caselet form, data is given in the form of reading comprehension or paragraph. To solve the question, data from comprehension is to be converted in the form of a Table or any other easy representation method.
Example 5: Directions: Study the following information carefully and answer the questions given beside.
Shana visited a shopping centre. She purchased some chocolates of three different brands, namely A, B, and C. The number of chocolates of each brand bought by her was equal to the price of the single chocolate of that particular brand. She has spent a sum of Rs. 602. If she had not bought brand C chocolates, then she would have spent only Rs. 313. Suppose, she purchases brand A chocolates at the price of brand B chocolates, brand B chocolates at the price of brand C chocolate, and the brand C chocolates at the price of brand A chocolates but the number of each brand of chocolates purchased by her remains the same then she spends Rs. 21 less than the price of all the chocolates for the actual price.
Q. What is the average number of chocolates purchased by Shana?
Sol: Let the price of single chocolate of A, B, and C brand chocolate is x, y, and z respectively
Then, according to the question, the number of chocolates of brands A, B, and C that were bought by her are x, y, and z respectively
As we know, that number of chocolates of each brand bought by her was equal to the price of single chocolate of that particular brand, and thus, she spent Rs. 602.
Accordingly, we have
The price of all the chocolates = x ∗ x + y ∗ y + z ∗ z = 602........... (i)
Further, it is known that if she had not bought brand C chocolates, then she would have spent only Rs. 313. Accordingly, we have, x ∗ x + y ∗ y = 313.............(ii)
Equation (i) – (ii)
z ∗z = 289
z = 17
Suppose, she purchases brand A chocolates at the price of brand B chocolates, brand B chocolates at the price of brand C chocolates, and the brand C chocolates at the price of brand A chocolates but the number of each brand of chocolates purchased by her remains the same then
x ∗ y + y ∗ z + z ∗ x = 602 – 21 = 581
We know than, (x + y + z)^{2} = x^{2} + y^{2} + z^{2} + 2(xy + yz +z x)= 602 + 2 × 581 = 1764
(x+y+z) 2 =1764
x + y + z = 42
x + y = 42 – 17 = 25_______ (iii)_____ (as z = 17)
x^{2} + y^{2} + 2xy = (x + y)^{2}
313 + 2xy = 625
2xy = 312
(x–y)^{2} = (x + y)^{2} – 4xy = 625 – 624 = 1
x – y =1 ______(iv)
By solving (iii) and (iv)
x = 13, y = 12
Thus, the average number of chocolates purchased by Shana be
56 videos104 docs95 tests

1. What is the difference between a point, a line segment, a ray, and a line? 
2. How are angles formed and measured? 
3. What does it mean for lines to be parallel? 
4. How do you classify triangles based on their angles? 
5. What are some common types of quadrilaterals? 
56 videos104 docs95 tests


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