In our day-to-day lives, we come across many situations like the temperature of a patient taken at different times of the day, sales of a shopkeeper on various days of the week, etc. In this chapter, we will discuss the construction and reading of graphs. In order to draw graphs, we first plot various points, and then they are joined by a freehand curve. Students who wish to have strong command over the subject can refer to RD Sharma Solutions for Class 8 Chapter 27 Introduction to Graphs. This resource can be used by the students to prepare for their exams as well. From the exam point of view, the solutions are designed by our experts in simple language, which helps in a better understanding of concepts. The solutions of RD Sharma Class 8 are available in PDF, which can be downloaded from the links given below.
Chapter 27, Introduction to Graphs, contains two exercises, and the RD Sharma Class 8 Solutions available on this page provide solutions to the questions present in each exercise. Now, let us have a look at the concepts discussed in this chapter.
- Cartesian plane
- Plotting of points
- Construction of graphs
RD Sharma Solutions for Class 8 Maths Chapter 27 Introduction to Graphs
Access RD Sharma Solutions for Class 8 Maths Chapter 27 Introduction to Graphs
EXERCISE 27.1 PAGE NO: 27.5
1. Plot the points (5, 0), (5, 1), and (5, 8). Do they lie on a line? What is your observation?
Solution:
Take point O on a graph paper and draw horizontal and vertical lines OX and OY, respectively.
Then, let on the x-axis and y-axis, 1 cm represents 1 unit.
To plot the point (5, 0), we start from the origin O and move 5 cm along X-axis. The point we arrive at is point (5, 0).
To plot the point (5, 1), we move 5 cm along X-axis and 1 cm along Y-axis. The point we arrive at is point (5, 1).
To plot the point (5, 8), we move 5 cm along X-axis and 8 cm along Y-axis. The point we arrive at is point (5, 8).
From the above graph, we observe that all points are having same X-coordinates, it can be seen that the points lie on a line parallel to the y-axis. Hence all points lie on the same line.
2. Plot the points (2, 8), (7, 8) and (12, 8). Join these points in pairs. Do they lie on a line? What do you observe?
Solution:
Take point O on a graph paper and draw the horizontal and vertical lines OX and OY, respectively.
Then, let on the x-axis and y-axis, 1 cm represents 1 unit.
In order to plot the point (2, 8), we start from the origin O and move 8 cm along X-axis. The point we arrive at is (2, 8).
To plot the point (7, 8), we move 7 cm along X-axis and 8 cm along Y-axis. The point we arrive at is (7, 8).
To plot the point (12, 8), we move 12 cm along X-axis and 8 cm along Y-axis. The point we arrive at is (12, 8).
From the above graph, we observe that all points have the same Y-coordinates, it can be seen that the points lie on a line parallel to the x-axis. Hence all points lie on the same line.
3. Locate the points :
(i) (1, 1), (1, 2), (1, 3), (1, 4)
(ii) (2, 1), (2, 2), (2, 3), (2, 4)
(iii) (1, 3), (2, 3), (3, 3), (4, 3)
(iv) (1, 4), (2, 4), (3, 4), (4, 4,)
Solution:
(i) (1, 1), (1, 2), (1, 3), (1, 4)
To plot these points,
Take point O on a graph paper and draw horizontal and vertical lines OX and OY, respectively.
Then, let on the x-axis, and y-axis 1 cm represents 1 unit.
To plot the point (1, 1), we start from the origin O and move 1 cm along X-axis and 1 cm along Y-axis. The point we arrive at is (1, 1).
To plot the point (1, 2), we move 1 cm along X-axis and 2 cm along Y-axis. The point we arrive at is (1, 2).
To plot the point (1, 3), we move 1 cm along X-axis and 3 cm along Y-axis. The point we arrive at is (1, 3).
To plot the point (1, 4), we move 1 cm along X-axis and 4 cm along Y-axis. The point we arrive at is (1, 4)
(ii) (2, 1), (2, 2), (2, 3), (2, 4)
To plot these points,
Take point O on a graph paper and draw horizontal and vertical lines OX and OY, respectively.
Then, let on the x-axis and y-axis, 1 cm represents 1 unit.
To plot the point (2, 1), we move 2 cm along X-axis and 1 cm along Y-axis. The point we arrive at is (2, 1).
To plot the point (2, 2), we move 2 cm along X-axis and 2 cm along Y-axis. The point we arrive at is (2, 2).
To plot the point (2, 3), we move 2 cm along X-axis and 3 cm along Y-axis. The point we arrive at is (2, 3).
To plot the point (2, 4), we move 2 cm along X-axis and 4 cm along Y-axis. The point we arrive at is (2, 4).
(iii) (1, 3), (2, 3), (3, 3), (4, 3)
To plot these points,
Take point O on a graph paper and draw horizontal and vertical lines OX and OY, respectively.
Then, let on the x-axis and y-axis, 1 cm represents 1 unit.
To plot the point (1, 3), we move 1 cm along X-axis and 3 cm along Y-axis. The point we arrive at is (1, 3).
To plot the point (2, 3), we move 2 cm along X-axis and 3 cm along Y-axis. The point we arrive at is (2, 3).
To plot the point (3, 3), we move 3 cm along X-axis and 3 cm along Y-axis. The point we arrive at is (3, 3).
To plot the point (4, 3), we move 4 0cm along X-axis and 3 cm along Y-axis. The point we arrive at is (4, 3).
(iv) (1, 4), (2, 4), (3, 4), (4, 4,)
To plot these points,
Take point O on a graph paper and draw horizontal and vertical lines OX and OY, respectively.
Then, let on the x-axis and y-axis, 1 cm represents 1 unit.
In order to plot the point (1, 4), we move 1 cm along X-axis and 4 cm along Y-axis. The point we arrive at is (1, 4).
To plot the point (2, 4), we move 2 cm along X-axis and 4 cm along Y-axis. The point we arrive at is (2, 4).
To plot the point (3, 4), we move 3 cm along X-axis and 4 cm along Y-axis. The point we arrive at is (3, 4).
To plot the point (4, 4), we move 4 cm along X-axis and 4 cm along Y-axis. The point we arrive at is (4, 4).
4. Find the coordinates of points A, B, C, D in Fig. 27.7
Solution:
Draw perpendiculars AP, BP, CQ and DR from A, B, C and D on the x-axis. Also, draw perpendiculars AW, BX, CY and DZ on the y-axis.
From the above figure, we have the following:
AW = 1 unit and AP= 1 unit
So, the coordinates of vertex A are (1, 1).
Similarly, BX=1 unit and BP= 4 units
So, the coordinates of vertex B are (1, 4).
CY = 4 units and CQ= 6 units
So, the coordinates of vertex C are (4, 6).
DZ = 5 units and DR= 3 units
So, the coordinates of vertex D are (5, 3).
5. Find the coordinates of points P, Q, R and S in Fig. 27.8.
Solution:
Draw perpendiculars PA, QB, RC and SD from vertices P, Q, R and S on the X-axis. Also, draw perpendiculars PE, QF, RG, and SH on the Y-axis from these points.
PE = 10 units and PA = 70 units
So, the coordinates of vertex P are (10, 70).
QF = 12 units and QB = 80 units
So, the coordinates of vertex Q are (12, 80).
RG = 16 units and RC = 100 units
So, the coordinates of vertex R are (16, 100).
SH = 20 units and SD = 120 units
So, the coordinates of vertex S are (20, 120).
6. Write the coordinates of each of the vertices of each polygon in Fig. 27.9.
Solution:
From the figure, we have the following:
In Quadrilateral OXYZ:
O lies on the origin, and the coordinates of the origin are (0, 0). So, the coordinates of O are (0, 0).
X lies on the Y-axis. So, the X – coordinate is 0. Hence, the coordinate of X is (0, 2).
Also, YX is equal to 2 units, and YZ is equal to 2 units. So, the coordinates of vertex Y are (2, 2).
Z lies on the X-axis. So, the Y-coordinate is 0. Hence, the coordinates of Z are (2, 0).
In polygon ABCD:
Draw perpendiculars DG, AH, CI and BJ from A, B, C and D on the X-axis.
Also, draw perpendiculars DF, AE, CF and BE from A, B, C and D on the Y-axis.
Now, from the figure:
DF = 3 units and DG = 3 units
So, the coordinates of D are (3, 3).
AE = 4 units and AH = 5 units
So, the coordinates of A are (4, 5).
CF = 6 units and CI = 3 units
So, the coordinates of C are (6, 3).
BE = 7 units and BJ = 5 units
So, the coordinates of B are (7, 5).
In polygon PQR:
Draw perpendiculars PJ, QK and RK from P, Q and R on the X-axis.
Also, draw perpendiculars PW, QE and RF from P, Q and R on the Y-axis.
Now, from the figure:
PW = 7 units and PJ = 4 units
So, the coordinates of P are (7, 4).
QE = 9 units and QK = 5 units
So, the coordinates of Q are (9, 5).
RF = 9 units and RK = 3 units
So, the coordinates of R are (9, 3)
7. Decide which of the following statements is true and which is false. Give reasons for your answer.
(i) A point whose x-coordinate is zero will lie on the y-axis.
(ii) A point whose y-coordinate is zero will lie on the x-axis.
(iii) The coordinates of the origin are (0, 0).
(iv) Points whose x and y coordinates are equal lie on a line passing through the origin.
Solution:
(i) A point whose x-coordinate is zero will lie on the y-axis.
From the figure,
For x = 0, we have x- coordinates as zero.
For example, (0, 3), (0, 6), (0, 9)
These points will lie on the y-axis. Hence, we say that our given statement is true.
(ii) A point whose y-coordinate is zero will lie on the x-axis.
A point whose y-coordinate is zero will lie on the x-axis.
For y = 0, we have y- coordinates as zero.
For example, (3, 0), (6, 0), (9, 0)
These points will lie on the x-axis. Hence, we say that our given statement is true.
(iii) The coordinates of the origin are (0, 0).
Origin is the intersection of the x-axis and y-axis. This means that the coordinates of the origin will be the intersection of lines y = 0 and x = 0.
Hence, the coordinates of the origin are (0, 0).
∴ The given statement is true.
(iv) Points whose x and y coordinates (0, 0), (1, 1), (2, 2) etc., are equal and lie on a line passing through the origin.
For the above statement, we can conclude that our statement satisfies the equation x = y.
For x = 0 and y = 0, this equation gets satisfied.
∴ The given statement is true.
EXERCISE 27.2 PAGE NO: 27.15
1. The following table shows the number of patients discharged from a hospital with HIV diagnosis in different years:
| Years | 2002 | 2003 | 2004 | 2005 | 2006 |
| Number of Patients | 150 | 170 | 195 | 225 | 230 |
Represent the information with a graph.
Solution:
Here we know that the year is an independent variable, and the number of patients is a dependent variable. So let us take years on the x-axis and the number of patients on the y-axis.
Let us choose the following scale:
On the x-axis: 2 cm = 1 year
On the y-axis: 1 cm = 10 patients
Let us assume that on the x-axis, origin (O) represents 2001, and on the y-axis, origin (O) represents 120, i.e. O (2001, 120).
Now, let us plot (2002, 150), (2003, 170), (2004, 195), (2005, 225), (2006, 230). These points are joined to get the graph representing the given information, as shown in the figure below.
2. The following table shows the amount of rice grown by a farmer in different years:
| Years | 2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 |
| Rice grown (in quintals) | 200 | 180 | 240 | 260 | 250 | 200 | 270 |
Plot a graph to illustrate this information.
Solution:
We know that, the year is an independent variable, and the quantity of rice grown is a dependent variable. So let us take years on the x-axis and the quantity of rice grown on the y-axis.
By choosing the following scale:
On the x-axis: 2 cm = 1 year
On the y-axis: 1 cm = 20 quintals
Let us assume that the origin O represents the coordinates (1999, 160).
Now, let us plot (2000, 200), (2001, 180), (2002, 240), (2003, 260), (2004, 250),(2005, 200),(2006, 270).
These points are joined to get the graph representing the given information, as shown in the figure below.
3. The following table gives information regarding the number of persons employed for a piece of work and the time taken to complete the work:
| Number of persons | 2 | 4 | 6 | 8 |
| Time taken (in days) | 12 | 6 | 4 | 3 |
Plot the graph of this information.
Solution:
We know that the number of persons is an independent variable, and time taken is a dependent variable.
So, we take the number of persons on the x-axis and the time taken on the y-axis.
By choosing the following scale:
On the x-axis: 2 cm = 2 persons
On the y-axis: 2 cm = 2 days
Now, let us plot (2, 12), (4, 6), (6, 4), and (8, 3).
These points are joined to get the graph representing the given information, as shown in the figure below.
4. The following table gives information regarding the length of a side of a square and its area:
| Length of a side (in cm) | 1 | 2 | 3 | 4 | 5 |
| Area of square (in cm2) | 1 | 4 | 9 | 16 | 25 |
Draw a graph to illustrate this information.
Solution:
We know that the length of a side is an independent variable, and the area of the square is a dependent variable.
So, let us take the length of a side on the x-axis and the area of the square on the y-axis.
By choosing the following scale:
On the x-axis: 2 cm = 1 cm
On the y-axis: 1 cm = 2 cm2
Now, let us plot (1, 1), (2, 4), (3, 9), (4, 16), and (5, 25).
These points are joined to get the graph representing the given information, as shown in the figure below.
5. The following table shows the sales of a commodity during the years 2000 to 2006.
| Years | 2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 |
| Sales (in lakhs of Rs) | 1.5 | 1.8 | 2.4 | 3.2 | 5.4 | 7.8 | 8.6 |
Draw a graph of this information.
Solution:
We know that the year is an independent variable, and sales is a dependent variable. So, we take the year on the x-axis and sales on the y-axis.
By choosing the following scale:
On the x-axis: 2 cm = 1 year
On the y-axis: 2 cm = 1 lakh rupees
Assume that on the x-axis, origin (O) represents 1991.
So, the coordinates of O are (1991, 0).
Now, let us plot (2000, 1.5), (2001, 1.8), (2002, 2.4), (2003, 3.2), (2004, 5.4), (2005, 7.8) and (2006, 8.6).
These points are joined to get the graph representing the given information, as shown in the figure below.
6. Draw the temperature-time graph in each of the following cases:
(i)
| Time (in hours) | 7:00 | 9:00 | 11:00 | 13:00 | 15:00 | 17:00 | 19:00 | 21:00 |
| Temperature (in oF) | 100 | 101 | 104 | 102 | 100 | 99 | 100 | 98 |
(ii)
| Time (in hours) | 8:00 | 10:00 | 12:00 | 14:00 | 16:00 | 18:00 | 20:00 |
| Temperature (in oF) | 100 | 101 | 104 | 103 | 99 | 98 | 100 |
Solution:
(i) We know that, time is an independent variable and temperature is a dependent variable. So, we take time on the x-axis and temperature on the y-axis.
Let us choose the following scale:
On the x-axis: 1 cm = 1 hour
On the y-axis: 1 cm = 1°F
Let us assume that on the x-axis, the coordinate of origin is 6:00.
On the y-axis, the coordinate of origin is 94°F.
So, the coordinates of 0 are (6:00, 94).
Now, let us plot (7:00, 100), (9:00, 101), (11:00, 104)…(21:00, 98)
These points are joined to get the graphs representing the given information, as shown in the figures below.
(ii) We know that, time is an independent variable and temperature is a dependent variable. So, we take time on the x-axis and temperature on the y-axis.
Let us choose the following scale:
On the x-axis: 2 cm = 2 hours
On the y-axis: 1 cm = 1°F
Let us assume that on the x-axis, the coordinate of origin is 6:00.
On the y-axis, the coordinate of origin is 94°F.
So, the coordinates of 0 are (6:00, 94).
Now, let us plot (8:00, 100), (10:00, 101), (12:00, 104) ….. (20:00, 100).
These points are joined to get the graphs representing the given information, as shown in the figures below.
7. Draw the velocity-time graph from the following data:
| Time (in hours) | 7:00 | 8:00 | 9:00 | 10:00 | 11:00 | 12:00 | 13:00 | 14:00 |
| Speed (in km/hr) | 30 | 45 | 60 | 50 | 70 | 50 | 40 | 45 |
Solution:
We know that, time is an independent variable and speed is a dependent variable. So, we take time on the x-axis and speed on the y-axis.
By choosing the following scale:
On the x-axis: 2 big divisions = 1 hour
On the y-axis: 1 big division = 10 km/hr
Let us assume that on the x-axis, the coordinate of origin (O) is 7:00.
So, the coordinates of O are (7:00, 0).
Now, let us plot (7:00, 30), (8:00, 45), (9:00, 60), (10:00, 50), (11:00, 70), (12:00, 50), (13:00, 40), (14:00, 45).
These points are joined to get the graph representing the given information, as shown in the figure below.
8. The runs scored by a cricket team in the first 15 overs are given below:
| Overs: | I | II | III | IV | V | VI | VII | VIII | IX | X | XI | XII | XIII | XIV | XV |
| Runs: | 2 | 1 | 4 | 2 | 6 | 8 | 10 | 21 | 5 | 8 | 3 | 2 | 6 | 8 | 12 |
Draw the graph representing the above data in two different ways as a graph and as a bar chart.
Solution:
We know that over is an independent variable and run is a dependent variable. So, we take overs on the x-axis and runs on the y-axis.
By choosing the following scale:
On the x-axis: 1 cm = 1 over
On the y-axis: 1 cm = 2 runs
Now, let us plot (I, 2), (II, 1), (III, 4)… (XV, 12).
These points are joined to get the graph representing the given information, as shown in the figure below.
The bar graph is as follows:
9. The runs scored by two teams, A and B, in the first 10 overs are given below:
| Overs | I | II | III | IV | V | VI | VII | VIII | IX | X |
| Team A | 2 | 1 | 8 | 9 | 4 | 5 | 6 | 10 | 6 | 2 |
| Team B | 5 | 6 | 2 | 10 | 5 | 6 | 3 | 4 | 8 | 10 |
Draw a graph depicting the data, making the graphs on the same axes in each case in two different ways as a graph and as a bar chart.
Solution:
We know that over is an independent variable and run is a dependent variable. So, we take overs on the x-axis and runs on the y-axis.
By choosing the following scale:
On the x-axis: 1 cm = 1 over
On the y-axis: 1 cm = 1 run
Now, let us plot (I, 2), (II, 1), (III, 8)… (X, 2) for team A and (I, 5), (II, 6), (III, 8) ….. (X, 10) for team B.
These points are joined to get the graph representing the given information, as shown in the figure below.
The bar graph is as follows:
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