Introduction To Graphs
Class 8th Mathematics RD Sharma Solution
- Plot the points (5,0), (5,1), (5,8). Do thy lie on a line? What is your…
- Plot the points (2,8), (7,8) and (12,8). Join these points in pairs. Do they…
- Locate the points : (i) (1,1), (1,2), (1,3), (1,4) (ii) (2,1), (2,2), (2,3),…
- Find the coordinates of points A, B, C, D in Fig. 27.7
- Find the coordinates of points P, Q, R and S in Fig. 27.8.
- Write the coordinates of each of the vertices of each polygon in Fig. 27.9.…
- Decide which of the following statements is true and which is false. Give…
- The following table shows the number of patients discharged from a hospital…
- The following table shows the amount of rice grown by a farmer in different…
- The following table gives the information regarding the number of persons…
- The following table gives the information regarding length of a side of a…
- The following table shows the sales of a commodity during the years 2000 to…
- Draw the temperature-time graph in each of the following cases: (i) Time (in…
- Draw the velocity-time graph from the following data : Time (in hours) 7:00…
- The runs scored by a cricket team in first 15 overs are given below : Overs : I…
- The runs scored by two teams A and B in first 10 overs are given below : Overs…
Exercise 27.1
Question 1.Plot the points (5,0), (5,1), (5,8). Do thy lie on a line? What is your observation?
Answer:
In order to plot point (5, 0), we start from the origin and move 5 cm along x- axis. The point we arrive at is point (5,0).
To plot 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 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).
After plotting points, we observe that all points are having same ordinate ( ie. same x-coordinate.)
Hence, all points lie on the same line.
Question 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?
Answer:
In order to plot point (2, 8), we move 2 cm along x-axis and 8 cm along y-axis. The point we arrive at is (2, 8).
To plot 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 point (12, 8), we move 12 cm along x-axis and 8 cm along y-axis. The point we arrive at is (12,8).
After plotting points, we observe that all points are having same ordinate ( ie. same y-coordinate.)
Hence, all points lie on the same line.
Question 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,)
Answer:
(i) (1,1), (1,2), (1,3), (1,4)
In order to plot point (1, 1), we move 1 cm along x-axis and 1 cm along y-axis. The point we arrive at is (1, 1).
To plot 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 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 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)
In order to plot 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 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 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 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)
In order to plot 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 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 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 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,)
In order to plot 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 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 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 point (4, 4), we move 4 0cm along x-axis and 4 cm along y-axis. The point we arrive at is (4,4).
Question 4.
Find the coordinates of points A, B, C, D in Fig. 27.7
Answer:
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 figure, we have:
AW = 1 cm and AP= 1 cm
So, the coordinates of vertex A are (1, 1).
Similarly, BX = 1 cm and BP = 4 cm
So, the coordinates of vertex B are (1, 4).
CY = 4 cm and CQ= 6 cm
So, the coordinates of vertex C are (4, 6).
DZ = 5 cm and DR= 3 cm
So, the coordinates of vertex D are (5, 3).
Question 5.
Find the coordinates of points P, Q, R and S in Fig. 27.8.
Answer:
Draw perpendiculars PE, QH, RI and SK from P, Q, C,R and S on the x-axis. Also, draw perpendiculars PF, QG, RJ and SL on the y-axis.
From the figure, we have:
PF = 10 cm and PE= 70 cm
So, the coordinates of vertex P are (10, 70).
Similarly, QG=12 cm and QH= 80 cm
So, the coordinates of vertex Q are (12, 80).
RJ = 16 cm and RI= 100 cm
So, the coordinates of vertex R are (4, 6).
SL = 20 cm and SK= 120 cm
So, the coordinates of vertex S are (5, 3).
Question 6.
Write the coordinates of each of the vertices of each polygon in Fig. 27.9.
Answer:
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 cm and YZ is equal to 2 cm. 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 and DF, AE, CF and BE from A, B, C and D on the y-axis.
DF = 3 cm and DG = 3 cm
Therefore, the coordinates of D are (3, 3).
AE = 4 cm and AH = 5 cm
Therefore, the coordinates of A are (4, 5).
CF = 6 cm and CI = 3 cm
Therefore, the coordinates of C are (6, 3).
BE = 7 cm and BJ = 5 cm
Therefore, the coordinates of B are (7, 5).
In triangle PQR,
Draw perpendiculars PJ, QK and RK from P, Q and R on the x-axis and PW, QE and RF from P, Q and R on the y-axis.
PW = 7 cm and PJ = 4 cm
Therefore, the coordinates of P are (7, 4).
QE = 9 cm and QK = 5 cm
Therefore, the coordinates of Q are (9, 5).
RF = 9 cm and RK = 3 cm
Therefore, the coordinates of R are (9, 3).
Question 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 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.
Answer:
For x = 0, we have x- coordinates as zero.
For example (0,5) and (0, 7)
These points will lie on y axis. Hence, we say that our given statement is true.
(ii) A point whose y-coordinate is zero, will lie on x-axis.
For y = 0, we have y- coordinates as zero.
For example (5,0) and (7, 0)
These points will lie on x axis. Hence, we say that our given statement is true.
(iii) The coordinates of the origin are (0,0).
Origin is intersection of x-axis and y-axis. This means that coordinates of the origin will be intersection of lines y = 0 and x = 0.
Hence, coordinates of origin are (0,0).
∴ Given statement is true.
(iv) Points whose x and y coordinates are equal, lie on a line passing through the origin.
For above statement we can conclude that our statement satisfies the equation x = y.
For x = 0 and y = 0, this equation gets satisfied.
Hence, our given statement is true.
Exercise 27.2
Question 1.The following table shows the number of patients discharged from a hospital with HIV diagnosis in different years:
Represent this information by a graph.
Answer:
Here, we take years on the x-axis and the number of patients on the y-axis.
Let us choose the following scale:
On x-axis: 1 cm = 1 year
On y-axis: 1 cm = 10 patients
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.
Scale:
On x axis: 1cm = 1year
On y-axis: 1 cm = 10 patients
Question 2.
The following table shows the amount of rice grown by a farmer in different years:
Plot a graph to illustrate this information.
Answer:
Here, we take years on the x-axis and quantity of rice grown on the y-axis.
Let us choose the following scale:
On x-axis: 1 cm = 1 year
On y-axis: 1 cm = 10 quintals
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.
Scale:
On x axis: 1cm = 1 year
On y-axis: 1 cm = 10 quintals
Question 3.
The following table gives the information regarding the number of persons employed to a piece of work and time taken to complete the work:
Plot a graph of this information.
Answer:
Here, we take the number of persons on the x-axis and time taken on the y-axis.
Let us choose the following scale:
On x-axis: 1 cm = 1 person
On y-axis: 1 cm = 1 day
Now, let us plot (2, 12), (4, 6), (6, 4), (8, 3). These points are joined to get the graph representing the given information as shown in the figure below.
Scale:
On x axis: 1cm = 1 person
On y-axis: 1 cm = 1 day
Question 4.
The following table gives the information regarding length of a side of a square and its area
Draw a graph to illustrate this information.
Answer:
Here, we take length of a side on the x-axis and area of square on the y-axis.
Let us choose the following scale:
On x-axis: 1 cm = 0.5 cm
On y-axis: 1 cm = 2 cm2
Now we plot (1,1), (2,4), (3,9), (4,16), (5,25). These points are joined to get the graph representing the given information as shown in the figure below.
Scale:
On x axis: 1cm = 0.5 cm
On y-axis: 2 cm = 1 cm2
Question 5.
The following table shows the sales of a commodity during the years 2000 to 2006.
Draw a graph of this information.
Answer:
Here, we take year on the x-axis and sales on the y-axis.
Let us choose the following scale:
On x-axis: 1 cm = 1 year
On y-axis: 1 cm = 1 lakh rupees
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
Scale:
On x axis: 1cm = 1 year
On y-axis: 1 cm = 1 lakh rupees
Question 6.
Draw the temperature-time graph in each of the following cases:
(i)
(ii)
Answer:
(i) Here, we take time on the x-axis and temperature on the y-axis.
Let us choose the following scale:
On x-axis: 1 cm = 1 hour
On y-axis: 1 cm = 1 °F
Now, let us plot the given points. These points are joined to get the graphs representing the given information as shown in the figures below.
Scale:
On x axis: 1cm = 1 hour
On y-axis: 1 cm = 1 °F
(ii) Here, we take time on the x-axis and temperature on the y-axis.
Let us choose the following scale:
On x-axis: 1 cm = 2 hour
On y-axis: 1 cm = 1 °F
Now, let us plot the given points. These points are joined to get the graphs representing the given information as shown in the figures below.
Scale:
On x axis: 1cm = 2 hour
On y-axis: 1 cm = 1 °F
Question 7.
Draw the velocity-time graph from the following data :
Answer:
Here, we take time on the x-axis and speed on the y-axis.
Let us choose the following scale:
On x-axis: 1 cm = 1 hour
On y-axis: 1 cm= 5 km/hr
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.
Scale:
On x axis: 1cm = 1 hour
On y-axis: 1 cm = 5 km/hr
Question 8.
The runs scored by a cricket team in first 15 overs are given below :
Answer:
Here, we take overs on the x-axis and runs the on y-axis.
Let us choose the following scale:
On x-axis: 1 cm = 1 over
On y-axis: 1 cm = 2 runs
Now, let us plot our given points. These points are joined to get the graph representing the given information as shown in the figure below.
Scale:
On x axis: 1cm = 1 over
On y-axis: 1 cm = 2 runs
Now we draw bar graph.
Question 9.
The runs scored by two teams A and B in first 10 overs are given below :
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.
Answer:
we take overs on x-axis and runs on y-axis.
Let us choose the following scale:
On x-axis: 1 cm = 1 over
On y-axis: 1 cm = 2 run
Now, let us plot given points for team A and for team B. These points are joined to get the graph representing the given information as shown in the figure below.
Now we plot bar graph
