In the previous section we saw some solved examples which demonstrated the method of plotting points. In this section we will see some practical applications of the Cartesian plane.
Consider the rectangle ABCD in fig.31.18 below.
The following data is given:
• It is drawn on a Cartesian plane.
♦ The scale is: 1 unit = 1 cm for both the axes
• Length and width of the rectangle ABCD are 4 cm and 2 cm respectively.
• The coordinates of it's lower left corner A is (3,2)
• The length AB is parallel to the x-axis
■ We are asked to find the coordinates of the other three corners B, C and D
Solution:
1. To find the coordinates of B:
(i) Given that AB is parallel to the x-axis. So every point on AB will be at the same distance from the x-axis.
• So every point on AB will have the same y coordinate as A.
• The corner B is a point on AB. So B will have the same y coordinate as A.
• Thus we get: y coordinate of B is 2
(ii) Length of AB is given as 4 cm. So B is at a horizontal distance of 4 cm from A
• A is at a horizontal distance of 3 units from the y-axis.
• So B will be at a horizontal distance of (3+4) = 7 units from the y-axis.
• Thus we get: x coordinate of B is 7
(iii) From (i) and (ii) we get: The coordinates of B are: (7,2)
2. To find the coordinates of C:
(i) BC is parallel to the y-axis. So every point on BC will be at the same distance from the y-axis.
• So every point on BC will have the same x coordinate as B.
• The corner C is a point on BC.
• So C will have the same x coordinate as B.
• Thus we get: x coordinate of C is 7
(ii) Length of BC is given as 2 cm. So C is at a vertical distance of 2 cm from B
• B is at a vertical distance of 2 units from the x-axis.
• So C will be at a vertical distance of (2+2) = 4 units from the x-axis.
• Thus we get: y coordinate of C is 4
(iii) From (i) and (ii) we get: The coordinates of C are: (7,4)
3. To find the coordinates of D:
(i) AD is parallel to the y-axis. So every point on AD will be at the same distance from the y-axis.
• So every point on AD will have the same x coordinate as A.
• The corner D is a point on AD.
• So D will have the same x coordinate as A.
• Thus we get: x coordinate of D is 3
(ii) Length of AD is given as 2 cm. So D is at a vertical distance of 2 cm from A
• A is at a vertical distance of 2 units from the x-axis.
• So D will be at a vertical distance of (2+2) = 4 units from the x-axis.
• Thus we get: y coordinate of D is 4
(iii) From (i) and (ii) we get: The coordinates of D are: (3,4)
The newly obtained coordinates are shown in the fig.31.19 below:
• Consider the line AB
♦ It is parallel to the x-axis
♦ If extended towards the left, it will pass through the point 2 on the y-axis
• All points on AB will have the same y coordinate '2'
• Consider the line DC
♦ It is parallel to the x-axis
♦ If extended towards the left, it will pass through the point 4 on the y-axis
• All points on DC will have the same y coordinate '4'
■ From this we can write:
• If a line is parallel to the x-axis, and if (when extended) it passes through a point 'm' on the y-axis, all points on that line will have the y-coordinate as 'm'
• In fact, that line can be represented by the equation: y = m
Fact 2:
• Consider the line AD
♦ It is parallel to the y-axis
♦ If extended towards the bottom, it will pass through the point 3 on the x-axis
• All points on AD will have the same x coordinate '3'
• Consider the line BC
♦ It is parallel to the y-axis
♦ If extended towards the bottom, it will pass through the point 7 on the x-axis
• All points on BC will have the same x coordinate '7'
■ From this we can write:
• If a line is parallel to the y-axis, and if (when extended) it passes through a point 'n' on the x-axis, all points on that line will have the x-coordinate as 'n'
• In fact, that line can be represented by the equation: x = n
Some examples are shown in the fig.31.20 below:
1. The coordinates of all the corners are given. After some simple calculations, we will be able to find two facts:
• The length of both the rectangles are 4 units
• The width of both the rectangles are 2 units
2. Now consider the fig.31.22 below:
• The position of 'O' has been changed. The origin is now at the lower left corner of the green rectangle. This is also the upper left corner of the blue rectangle
3. After some simple calculations, we will be able to find two facts:
• The length of both the rectangles are 4 units
• The width of both the rectangles are 2 units
The reader may do the calculations in his/her own notebooks and verify the above facts
4. So we find that, changing the position of the coordinate axes do not have any effect on the properties of the 'objects'. The only change is to the 'coordinates of points'.
5. That means, we can place the axes at any convenient point that we like. But the following conditions should be satisfied:
• The axes must be perpendicular to each other
• Once the position of axes is fixed, it must not be changed
Find the coordinates of the other three vertices of the rectangle OABC in the fig.31.23(a) below
Solution:
1. Coordinates of 'O' are obviously (0,0)
2. Coordinates of A:
(i) Given that OABC is a rectangle.
• So AB is parallel to OC.
• But OC lies on the y-axis. So AB is parallel to the y-axis
• So any point on AB will have the same x coordinate as B
• Thus the x coordinate of A is 4
(ii) A lies on the x-axis
• Any point on the x-axis will have the y coordinate as '0'
(iii) From (i) and (ii) we get: coordinates of A are (4,0)
3. Coordinates of C:
(i) Given that OABC is a rectangle.
• So BC is parallel to OA.
• But OA lies on the x-axis. So BC is parallel to the x-axis
• So any point on BC will have the same y coordinate as B
• Thus the y coordinate of C is 3
(ii) C lies on the y-axis
• Any point on the y-axis will have the x coordinate as '0'
(iii) From (i) and (ii) we get: coordinates of C are (0,3)
• The newly calculated coordinates are shown in fig.31.23(b) above
Solved example 31.9
Find the coordinates of the other three vertices A, B and D of the rectangle in the fig.31.24(a) below. Given that, the origin O is at the centre of the rectangle and the sides of the rectangle are parallel to the axes.
Solution:
1. Coordinates of B:
(i) Given that BC is parallel to y-axis
• So any point on BC will have the same x coordinate as B
• Thus the x coordinate of B is 3
(ii) The x-axis will pass through the midpoint of BC. So we get:
• Distance of C from x-axis = Distance of B from x-axis
• But distance of C from x-axis = it's y coordinate = 2
• So distance of B from the x axis = 2
• But B is in the fourth quadrant. In the fourth quadrant, the y-coordinate should be given -ve sign
• So the y coordinate of B is -2
(iii) From (i) and (ii) we get: coordinates of B are (3,-2)
2. Coordinates of A:
(i) AB is parallel to x-axis
• So any point on AB will have the same y coordinate as B
• Thus the y coordinate of A is -2
(ii) The y-axis will pass through the midpoint of AB. So we get:
• Distance of B from y-axis = Distance of A from y-axis
• But distance of B from y-axis = it's x coordinate = 3
• So distance of A from the x axis = 3
• But A is in the third quadrant. In the third quadrant, the x-coordinate should be given -ve sign
• So the x coordinate of A is -3
(iii) From (i) and (ii) we get: coordinates of A are (-3,-2)
3. Coordinates of D:
(i) CD is parallel to x-axis
• So any point on CD will have the same y coordinate as C
• Thus the y coordinate of D is 2
(ii) The y-axis will pass through the midpoint of CD. So we get:
• Distance of C from y-axis = Distance of D from y-axis
• But distance of C from y-axis = it's x coordinate = 3
• So distance of D from the x axis = 3
• But D is in the second quadrant. In the second quadrant, the x-coordinate should be given -ve sign
• So the x coordinate of D is -3
(iii) From (i) and (ii) we get: coordinates of D are (-3,2)
Solved example 31.10
The isosceles triangle ABC shown in fig.31.25(a) has the base AB of 3 units and height 4 units.
What are the coordinates of the vertices A, B and C?
Solution:
1. The x-axis is passing through the base
• The y-axis, which is always perpendicular to the x-axis, is passing through the apex
• So the altitude of the given triangle lies on the y-axis
• Thus we get: Distance of C from the x-axis = altitude of the triangle = 4 units
• So y coordinate of C = 4 (This is +ve 4 because it is on OY, the positive side of y-axis)
2. The x coordinate of any point on the y axis is 0
3. From (1) and (2) we get: Coordinates of C are: (0,4)
4. Since y axis is the altitude, it bisects the base AB perpendicularly. So OA = OB = 3⁄2 = 1.5 units
• Now we can write the coordinates of A and B:
5. Coordinates of A:
(i) A is at a distance of 1.5 units from y-axis. So the x coordinate of A is -1.5 (-ve sign is given because A lies on OX' which is the negative side of the x-axis)
(ii) The y coordinate of any point on the x-axis is 0
(iii) From (i) and (ii) we get: Coordinates of A are (-1.5,0)
6. Coordinates of B:
(i) B is at a distance of 1.5 units from y-axis. So the x coordinate of B is +1.5 (+ve sign is given because B lies on OX which is the positive side of the x-axis)
(ii) The y coordinate of any point on the x-axis is 0
(iii) From (i) and (ii) we get: Coordinates of B are (1.5,0)
7. The results are shown in fig,31.25(b)
Solved example 31.11
An equilateral triangle OAB having side 4 units is placed in the Cartesian plane as shown in the fig.31.26(a) below:
Find the coordinates of the vertices A and B
Solution:
1. Coordinates of A:
(i) A lies on the x-axis and is at a distance of 4 units from O
• So it's x coordinate is +4 (∵ it lies on the positive side OX)
(ii) All points on the x-axis will have y coordinate as 0.
(iii) From (i) and (ii) we get: coordinates of A are (4,0)
2. Coordinates of B:
(i) Consider fig.31.26(b). Two white dashed lines, one horizontal and the other vertical, pass through B
(ii) We want the following two information:
• The distance of the horizontal dashed line from the x-axis
• The distance of the vertical dashed line from the y-axis
(iii) First consider the vertical dashed line.
• Since the triangle is equilateral, the vertical dashed line will pass through the midpoint of the base OA
• That means, the vertical dashed line is at a distance of 2 units from the y-axis
• x coordinate of any point on this vertical dashed line will be 2
• So x coordinate of B is 2
(iv) Now consider the horizontal dashed line.
• The distance of this line from the x-axis is same as the height of the given equilateral triangle
• We have: Height of any equilateral triangle of side s = (√3)s⁄2 (Details here)
• So the height of our equilateral triangle OAB = (√3)× 4⁄2 = 2√3
• So distance of the horizontal dashed line from the x-axis is 2√3
• y coordinate of any point on this horizontal dashed line will be 2√3
• So y coordinate of B is 2√3
(v) From (iii) and (iv) we get: Coordinates of B are: (2,2√3)
Solved example 31.12
A parallelogram having longer side 6 units and shorter side 4 units is placed in the Cartesian plane as shown in the fig.31.27(a) below:
The included angle is 60o. Find the coordinates of the vertices A, B and C
Solution:
1. Coordinates of A:
(i) A lies on the x-axis and is at a distance of 6 units from O
• So it's x coordinate is +6 (∵ it lies on the positive side OX)
(ii) All points on the x-axis will have y coordinate as 0.
(iii) From (i) and (ii) we get: coordinates of A are (6,0)
2. Coordinates of B:
(i) Consider fig.31.27(b). Two white dashed lines, one horizontal and the other vertical, pass through B
(ii) We want the following two information:
• The distance of the horizontal dashed line from the x-axis
• The distance of the vertical dashed line from the y-axis
(iii) First consider the vertical dashed line.
It meets the x-axis at B'. So we have a right triangle AB'B
• It is a 30o, 60o triangle. We have seen the details of such triangles here.
• The hypotenuse will be 2 times the smallest side
• So AB = 2×AB' ⟹AB' = AB⁄2 = 4⁄2 = 2 units
• Then distance OB' = (OA + AB') = (6+2) = 8 units
• That means, the vertical dashed line is at a distance of 8 units from the y-axis
• x coordinate of any point on this vertical dashed line will be 8
• So x coordinate of B is 8
(iv) Now consider the horizontal dashed line
• In the right triangle AB'B, the altitude will be √3 times the smallest side
• So BB' = √3×AB' = 2√3
• So distance of the horizontal dashed line from the x-axis is 2√3
• y coordinate of any point on this horizontal dashed line will be 2√3
• So y coordinate of B is 2√3
(v) From (iii) and (iv) we get: Coordinates of B are: (8,2√3)
3. Coordinates of C:
(i) C is on the same horizontal line as B.
• So the y coordinate of C will be same as that of B, which is 2√3
(ii) The horizontal dashed line meets the y-axis at C'
• By properties of a parallelogram, C'C will be equal to AB' which is 2 units
• So the x coordinate of C is 2
(iii) From (i) and (i) we get: Coordinates of C are: (2,2√3)
Solved example 31.13
In the fig.31.28(a) below, a large trapezium is made up of four equal trapeziums.
Find the coordinates of the vertices of all the trapeziums
Solution:
1. Consider the red and green trapeziums at the bottom
• Two equal bases give a distance of 8 units. So each of them is (8⁄2) = 4 units
2. Both the bases lie on the x-axis. So we get:
• Bottom right vertex of red is (4,0)
• Bottom right vertex of green is (8,0)
3. The height of the total large trapezium is the base of the magenta trapezium
• But base of magenta = base of red = 4 units
• So top right of magenta will have a y coordinate of 4
• It's x coordinate will be: (base of red + base of green) = (4+4) = 8
• So the coordinates at the top right of magenta is (8,4)
4. y coordinate at the top left of magenta will be same as that at it's top right which is 4
• Height of red + height of blue = total height = 4
• So height of red = height of blue = height of each = 4⁄2 = 2 units
• x coordinate of top left of magenta = (x coordinate of top right of magenta - 2) = (8-2) = 6
• So the top left of magenta is (6,4)
■ In this way all other vertices can be calculated. They are shown in fig.b
Note:
• Top of red + top of green = base of blue = 4
• So top of red = top of green = top of each = 4⁄2 = 2 units
Solved example 31.14
All the rectangles in fig.31.29 below have sides parallel to the axes.
Find the coordinates of the remaining vertices of each.
Solution:
Fig.a:
1. To find the coordinates of B:
(i) Given that AB is parallel to the x-axis. So every point on AB will be at the same distance from the x-axis.
• So every point on AB will have the same y coordinate as A.
• The corner B is a point on AB. So B will have the same y coordinate as A.
• Thus we get: y coordinate of B is 3
(ii) Given that BC is parallel to the y-axis. So every point on BC will be at the same distance from the y-axis.
• So every point on BC will have the same x coordinate as A.
• The corner B is a point on BC. So B will have the same x coordinate as C.
• Thus we get: x coordinate of B is 2
(iii) From (i) and (ii) we get: The coordinates of B are: (2,3)
2. To find the coordinates of D:
(i) Given that CD is parallel to the x-axis. So every point on CD will be at the same distance from the x-axis.
• So every point on CD will have the same y coordinate as C.
• The corner D is a point on CD. So D will have the same y coordinate as C.
• Thus we get: y coordinate of D is 4
(ii) Given that AD is parallel to the y-axis. So every point on AD will be at the same distance from the y-axis.
• So every point on AD will have the same x coordinate as A.
• The corner D is a point on AD. So D will have the same x coordinate as A.
• Thus we get: x coordinate of D is -2
(iii) From (i) and (ii) we get: The coordinates of B are: (-2,4)
Fig.b:
1. To find the coordinates of A:
(i) Given that AB is parallel to the x-axis. So every point on AB will be at the same distance from the x-axis.
• So every point on AB will have the same y coordinate as B.
• The corner A is a point on AB. So A will have the same y coordinate as B.
• Thus we get: y coordinate of A is -4
(ii) Given that AD is parallel to the y-axis. So every point on AD will be at the same distance from the y-axis.
• So every point on AD will have the same x coordinate as D.
• The corner A is a point on AD. So A will have the same x coordinate as D.
• Thus we get: x coordinate of A is -1
(iii) From (i) and (ii) we get: The coordinates of B are: (-1,-4)
2. To find the coordinates of C:
(i) Given that CD is parallel to the x-axis. So every point on CD will be at the same distance from the x-axis.
• So every point on CD will have the same y coordinate as D.
• The corner C is a point on CD. So C will have the same y coordinate as D.
• Thus we get: y coordinate of C is -2
(ii) Given that BC is parallel to the y-axis. So every point on BC will be at the same distance from the y-axis.
• So every point on BC will have the same x coordinate as B.
• The corner C is a point on BC. So C will have the same x coordinate as B.
• Thus we get: x coordinate of C is 2
(iii) From (i) and (ii) we get: The coordinates of C are: (2,-2)
Fig.c:
1. To find the coordinates of B:
(i) Given that AB is parallel to the x-axis. So every point on AB will be at the same distance from the x-axis.
• So every point on AB will have the same y coordinate as A.
• The corner B is a point on AB. So B will have the same y coordinate as A.
• Thus we get: y coordinate of B is 3
(ii) Given that BC is parallel to the y-axis. So every point on BC will be at the same distance from the y-axis.
• So every point on BC will have the same x coordinate as A.
• The corner B is a point on BC. So B will have the same x coordinate as C.
• Thus we get: x coordinate of B is 2
(iii) From (i) and (ii) we get: The coordinates of B are: (2,3)
2. To find the coordinates of D:
(i) Given that CD is parallel to the x-axis. So every point on CD will be at the same distance from the x-axis.
• So every point on CD will have the same y coordinate as C.
• The corner D is a point on CD. So D will have the same y coordinate as C.
• Thus we get: y coordinate of D is 6
(ii) Given that AD is parallel to the y-axis. So every point on AD will be at the same distance from the y-axis.
• So every point on AD will have the same x coordinate as A.
• The corner D is a point on AD. So D will have the same x coordinate as A.
• Thus we get: x coordinate of D is -1
(iii) From (i) and (ii) we get: The coordinates of B are: (-1,6)
The completed fig.31.30 is shown below:
Solved example 31.15
Given below are four pairs of coordinates. Each pair contains coordinates of any two corners of rectangles whose sides are parallel to the axes. In each pair, mark each coordinate as top-left, top-right, bottom-left or bottom-right. Mark them with out drawing axes. After marking, find the other coordinates also.
(i) (3,5), (7,8)
(ii) (6,2), (5,4)
(iii) (-3,5), (-7,1)
(iv) (-1,-2), (-5,-4)
Solution:
Case (i):
1. Consider the x coordinates: 3 and 7
(i) 7 is greater than 3. So (7,8) is on the right side of (3,5)
(ii) Thus the possibilities for (3,5) are: left top or left bottom
(iii) Also possibilities for (7,8) are: right top or right bottom
2. Consider the y coordinates: 5 and 8
(i) 8 is greater than 5. So (3,5) is at bottom and (7,8) is at top
(ii) Thus the possibilities for (3,5) are: left bottom or right bottom
(iii) Also possibilities for (7,8) are: left top or right top
3. Combining 1(ii) and 2(ii) we get: (3,5) is left bottom [Taking the common possibility. See the green highlighted lines]
4. Combining 1(iii) and 2(iii) we get: (7,8) is right top [Taking the common possibility. See the red highlighted lines]
This is shown in the fig.31.31 below:
Case (ii):
1. Consider the x coordinates: 6 and 5
(i) 6 is greater than 5. So (6,2) is on the right side of (5,4)
(ii) Thus the possibilities for (5,4) are: left top or left bottom
(iii) Also possibilities for (6,2) are: right top or right bottom
2. Consider the y coordinates: 2 and 4
(i) 5 is greater than 2. So (6,2) is at bottom and (5,4) is at top
(ii) Thus the possibilities for (6,2) are: left bottom or right bottom
(iii) Also possibilities for (5,4) are: left top or right top
3. Combining 1(ii) and 2(iii) we get: (5,4) is left top [Taking the common possibility]
4. Combining 1(iii) and 2(ii) we get: (6,2) is right bottom [Taking the common possibility]
This is shown in the fig. above
Case (iii):
1. Consider the x coordinates: -3 and -7
(i) -3 is greater than -7. So (-3,5) is on the right side of (-7,1)
(ii) Thus the possibilities for (-7,1) are: left top or left bottom
(iii) Also possibilities for (-3,-5) are: right top or right bottom
2. Consider the y coordinates: 5 and 1
(i) 5 is greater than 1. So (-7,1) is at bottom and (-3,5) is at top
(ii) Thus the possibilities for (-7,1) are: left bottom or right bottom
(iii) Also possibilities for (-3,5) are: left top or right top
3. Combining 1(ii) and 2(ii) we get: (-7,1) is left bottom [Taking the common possibility]
4. Combining 1(iii) and 2(ii) we get: (-3,5) is right top [Taking the common possibility]
This is shown in the fig. above
Case (iv):
1. Consider the x coordinates: -1 and -5
(i) -1 is greater than -5. So (-1,-2) is on the right side of (-5,-4)
(ii) Thus the possibilities for (-5,4) are: left top or left bottom
(iii) Also possibilities for (-1,-2) are: right top or right bottom
2. Consider the y coordinates: -2 and -4
(i) -2 is greater than -4. So (-5,-4) is at bottom and (-1,-2) is at top
(ii) Thus the possibilities for (-5,-4) are: left bottom or right bottom
(iii) Also possibilities for (-1,-2) are: left top or right top
3. Combining 1(ii) and 2(ii) we get: (-5,-4) is left bottom [Taking the common possibility]
4. Combining 1(iii) and 2(ii) we get: (-1,-2) is right top [Taking the common possibility]
This is shown in the fig. above
Part 2:
The remaining coordinates of each rectangle can be calculated by using the procedure that we saw in the previous example. The completed figs are shown below:
Consider the rectangle ABCD in fig.31.18 below.
 |
| Fig.31.18 |
• It is drawn on a Cartesian plane.
♦ The scale is: 1 unit = 1 cm for both the axes
• Length and width of the rectangle ABCD are 4 cm and 2 cm respectively.
• The coordinates of it's lower left corner A is (3,2)
• The length AB is parallel to the x-axis
■ We are asked to find the coordinates of the other three corners B, C and D
Solution:
1. To find the coordinates of B:
(i) Given that AB is parallel to the x-axis. So every point on AB will be at the same distance from the x-axis.
• So every point on AB will have the same y coordinate as A.
• The corner B is a point on AB. So B will have the same y coordinate as A.
• Thus we get: y coordinate of B is 2
(ii) Length of AB is given as 4 cm. So B is at a horizontal distance of 4 cm from A
• A is at a horizontal distance of 3 units from the y-axis.
• So B will be at a horizontal distance of (3+4) = 7 units from the y-axis.
• Thus we get: x coordinate of B is 7
(iii) From (i) and (ii) we get: The coordinates of B are: (7,2)
2. To find the coordinates of C:
(i) BC is parallel to the y-axis. So every point on BC will be at the same distance from the y-axis.
• So every point on BC will have the same x coordinate as B.
• The corner C is a point on BC.
• So C will have the same x coordinate as B.
• Thus we get: x coordinate of C is 7
(ii) Length of BC is given as 2 cm. So C is at a vertical distance of 2 cm from B
• B is at a vertical distance of 2 units from the x-axis.
• So C will be at a vertical distance of (2+2) = 4 units from the x-axis.
• Thus we get: y coordinate of C is 4
(iii) From (i) and (ii) we get: The coordinates of C are: (7,4)
3. To find the coordinates of D:
(i) AD is parallel to the y-axis. So every point on AD will be at the same distance from the y-axis.
• So every point on AD will have the same x coordinate as A.
• The corner D is a point on AD.
• So D will have the same x coordinate as A.
• Thus we get: x coordinate of D is 3
(ii) Length of AD is given as 2 cm. So D is at a vertical distance of 2 cm from A
• A is at a vertical distance of 2 units from the x-axis.
• So D will be at a vertical distance of (2+2) = 4 units from the x-axis.
• Thus we get: y coordinate of D is 4
(iii) From (i) and (ii) we get: The coordinates of D are: (3,4)
The newly obtained coordinates are shown in the fig.31.19 below:
 |
| Fig.31.19 |
We can see two interesting facts from the above problem:
Fact 1:• Consider the line AB
♦ It is parallel to the x-axis
♦ If extended towards the left, it will pass through the point 2 on the y-axis
• All points on AB will have the same y coordinate '2'
• Consider the line DC
♦ It is parallel to the x-axis
♦ If extended towards the left, it will pass through the point 4 on the y-axis
• All points on DC will have the same y coordinate '4'
■ From this we can write:
• If a line is parallel to the x-axis, and if (when extended) it passes through a point 'm' on the y-axis, all points on that line will have the y-coordinate as 'm'
• In fact, that line can be represented by the equation: y = m
Fact 2:
• Consider the line AD
♦ It is parallel to the y-axis
♦ If extended towards the bottom, it will pass through the point 3 on the x-axis
• All points on AD will have the same x coordinate '3'
• Consider the line BC
♦ It is parallel to the y-axis
♦ If extended towards the bottom, it will pass through the point 7 on the x-axis
• All points on BC will have the same x coordinate '7'
■ From this we can write:
• If a line is parallel to the y-axis, and if (when extended) it passes through a point 'n' on the x-axis, all points on that line will have the x-coordinate as 'n'
• In fact, that line can be represented by the equation: x = n
Some examples are shown in the fig.31.20 below:
 |
| Fig.31.20 |
Position of the axes
Consider the blue and green rectangles in fig.31.21 below: |
| Fig.31.21 |
• The length of both the rectangles are 4 units
• The width of both the rectangles are 2 units
2. Now consider the fig.31.22 below:
 |
| Fig.31.22 |
3. After some simple calculations, we will be able to find two facts:
• The length of both the rectangles are 4 units
• The width of both the rectangles are 2 units
The reader may do the calculations in his/her own notebooks and verify the above facts
4. So we find that, changing the position of the coordinate axes do not have any effect on the properties of the 'objects'. The only change is to the 'coordinates of points'.
5. That means, we can place the axes at any convenient point that we like. But the following conditions should be satisfied:
• The axes must be perpendicular to each other
• Once the position of axes is fixed, it must not be changed
Now we will see some solved examples:
Solved example 31.8Find the coordinates of the other three vertices of the rectangle OABC in the fig.31.23(a) below
 |
| Fig.31.23 |
1. Coordinates of 'O' are obviously (0,0)
2. Coordinates of A:
(i) Given that OABC is a rectangle.
• So AB is parallel to OC.
• But OC lies on the y-axis. So AB is parallel to the y-axis
• So any point on AB will have the same x coordinate as B
• Thus the x coordinate of A is 4
(ii) A lies on the x-axis
• Any point on the x-axis will have the y coordinate as '0'
(iii) From (i) and (ii) we get: coordinates of A are (4,0)
3. Coordinates of C:
(i) Given that OABC is a rectangle.
• So BC is parallel to OA.
• But OA lies on the x-axis. So BC is parallel to the x-axis
• So any point on BC will have the same y coordinate as B
• Thus the y coordinate of C is 3
(ii) C lies on the y-axis
• Any point on the y-axis will have the x coordinate as '0'
(iii) From (i) and (ii) we get: coordinates of C are (0,3)
• The newly calculated coordinates are shown in fig.31.23(b) above
Solved example 31.9
Find the coordinates of the other three vertices A, B and D of the rectangle in the fig.31.24(a) below. Given that, the origin O is at the centre of the rectangle and the sides of the rectangle are parallel to the axes.
 |
| Fig.31.24 |
1. Coordinates of B:
(i) Given that BC is parallel to y-axis
• So any point on BC will have the same x coordinate as B
• Thus the x coordinate of B is 3
(ii) The x-axis will pass through the midpoint of BC. So we get:
• Distance of C from x-axis = Distance of B from x-axis
• But distance of C from x-axis = it's y coordinate = 2
• So distance of B from the x axis = 2
• But B is in the fourth quadrant. In the fourth quadrant, the y-coordinate should be given -ve sign
• So the y coordinate of B is -2
(iii) From (i) and (ii) we get: coordinates of B are (3,-2)
2. Coordinates of A:
(i) AB is parallel to x-axis
• So any point on AB will have the same y coordinate as B
• Thus the y coordinate of A is -2
(ii) The y-axis will pass through the midpoint of AB. So we get:
• Distance of B from y-axis = Distance of A from y-axis
• But distance of B from y-axis = it's x coordinate = 3
• So distance of A from the x axis = 3
• But A is in the third quadrant. In the third quadrant, the x-coordinate should be given -ve sign
• So the x coordinate of A is -3
(iii) From (i) and (ii) we get: coordinates of A are (-3,-2)
3. Coordinates of D:
(i) CD is parallel to x-axis
• So any point on CD will have the same y coordinate as C
• Thus the y coordinate of D is 2
(ii) The y-axis will pass through the midpoint of CD. So we get:
• Distance of C from y-axis = Distance of D from y-axis
• But distance of C from y-axis = it's x coordinate = 3
• So distance of D from the x axis = 3
• But D is in the second quadrant. In the second quadrant, the x-coordinate should be given -ve sign
• So the x coordinate of D is -3
(iii) From (i) and (ii) we get: coordinates of D are (-3,2)
Solved example 31.10
The isosceles triangle ABC shown in fig.31.25(a) has the base AB of 3 units and height 4 units.
 |
| Fig.31.25 |
Solution:
1. The x-axis is passing through the base
• The y-axis, which is always perpendicular to the x-axis, is passing through the apex
• So the altitude of the given triangle lies on the y-axis
• Thus we get: Distance of C from the x-axis = altitude of the triangle = 4 units
• So y coordinate of C = 4 (This is +ve 4 because it is on OY, the positive side of y-axis)
2. The x coordinate of any point on the y axis is 0
3. From (1) and (2) we get: Coordinates of C are: (0,4)
4. Since y axis is the altitude, it bisects the base AB perpendicularly. So OA = OB = 3⁄2 = 1.5 units
• Now we can write the coordinates of A and B:
5. Coordinates of A:
(i) A is at a distance of 1.5 units from y-axis. So the x coordinate of A is -1.5 (-ve sign is given because A lies on OX' which is the negative side of the x-axis)
(ii) The y coordinate of any point on the x-axis is 0
(iii) From (i) and (ii) we get: Coordinates of A are (-1.5,0)
6. Coordinates of B:
(i) B is at a distance of 1.5 units from y-axis. So the x coordinate of B is +1.5 (+ve sign is given because B lies on OX which is the positive side of the x-axis)
(ii) The y coordinate of any point on the x-axis is 0
(iii) From (i) and (ii) we get: Coordinates of B are (1.5,0)
7. The results are shown in fig,31.25(b)
Solved example 31.11
An equilateral triangle OAB having side 4 units is placed in the Cartesian plane as shown in the fig.31.26(a) below:
 |
| Fig.31.26 |
Solution:
1. Coordinates of A:
(i) A lies on the x-axis and is at a distance of 4 units from O
• So it's x coordinate is +4 (∵ it lies on the positive side OX)
(ii) All points on the x-axis will have y coordinate as 0.
(iii) From (i) and (ii) we get: coordinates of A are (4,0)
2. Coordinates of B:
(i) Consider fig.31.26(b). Two white dashed lines, one horizontal and the other vertical, pass through B
(ii) We want the following two information:
• The distance of the horizontal dashed line from the x-axis
• The distance of the vertical dashed line from the y-axis
(iii) First consider the vertical dashed line.
• Since the triangle is equilateral, the vertical dashed line will pass through the midpoint of the base OA
• That means, the vertical dashed line is at a distance of 2 units from the y-axis
• x coordinate of any point on this vertical dashed line will be 2
• So x coordinate of B is 2
(iv) Now consider the horizontal dashed line.
• The distance of this line from the x-axis is same as the height of the given equilateral triangle
• We have: Height of any equilateral triangle of side s = (√3)s⁄2 (Details here)
• So the height of our equilateral triangle OAB = (√3)× 4⁄2 = 2√3
• So distance of the horizontal dashed line from the x-axis is 2√3
• y coordinate of any point on this horizontal dashed line will be 2√3
• So y coordinate of B is 2√3
(v) From (iii) and (iv) we get: Coordinates of B are: (2,2√3)
Solved example 31.12
A parallelogram having longer side 6 units and shorter side 4 units is placed in the Cartesian plane as shown in the fig.31.27(a) below:
 |
| Fig.31.27 |
Solution:
1. Coordinates of A:
(i) A lies on the x-axis and is at a distance of 6 units from O
• So it's x coordinate is +6 (∵ it lies on the positive side OX)
(ii) All points on the x-axis will have y coordinate as 0.
(iii) From (i) and (ii) we get: coordinates of A are (6,0)
2. Coordinates of B:
(i) Consider fig.31.27(b). Two white dashed lines, one horizontal and the other vertical, pass through B
(ii) We want the following two information:
• The distance of the horizontal dashed line from the x-axis
• The distance of the vertical dashed line from the y-axis
(iii) First consider the vertical dashed line.
It meets the x-axis at B'. So we have a right triangle AB'B
• It is a 30o, 60o triangle. We have seen the details of such triangles here.
• The hypotenuse will be 2 times the smallest side
• So AB = 2×AB' ⟹AB' = AB⁄2 = 4⁄2 = 2 units
• Then distance OB' = (OA + AB') = (6+2) = 8 units
• That means, the vertical dashed line is at a distance of 8 units from the y-axis
• x coordinate of any point on this vertical dashed line will be 8
• So x coordinate of B is 8
(iv) Now consider the horizontal dashed line
• In the right triangle AB'B, the altitude will be √3 times the smallest side
• So BB' = √3×AB' = 2√3
• So distance of the horizontal dashed line from the x-axis is 2√3
• y coordinate of any point on this horizontal dashed line will be 2√3
• So y coordinate of B is 2√3
(v) From (iii) and (iv) we get: Coordinates of B are: (8,2√3)
3. Coordinates of C:
(i) C is on the same horizontal line as B.
• So the y coordinate of C will be same as that of B, which is 2√3
(ii) The horizontal dashed line meets the y-axis at C'
• By properties of a parallelogram, C'C will be equal to AB' which is 2 units
• So the x coordinate of C is 2
(iii) From (i) and (i) we get: Coordinates of C are: (2,2√3)
Solved example 31.13
In the fig.31.28(a) below, a large trapezium is made up of four equal trapeziums.
 |
| Fig.31.28 |
Solution:
1. Consider the red and green trapeziums at the bottom
• Two equal bases give a distance of 8 units. So each of them is (8⁄2) = 4 units
2. Both the bases lie on the x-axis. So we get:
• Bottom right vertex of red is (4,0)
• Bottom right vertex of green is (8,0)
3. The height of the total large trapezium is the base of the magenta trapezium
• But base of magenta = base of red = 4 units
• So top right of magenta will have a y coordinate of 4
• It's x coordinate will be: (base of red + base of green) = (4+4) = 8
• So the coordinates at the top right of magenta is (8,4)
4. y coordinate at the top left of magenta will be same as that at it's top right which is 4
• Height of red + height of blue = total height = 4
• So height of red = height of blue = height of each = 4⁄2 = 2 units
• x coordinate of top left of magenta = (x coordinate of top right of magenta - 2) = (8-2) = 6
• So the top left of magenta is (6,4)
■ In this way all other vertices can be calculated. They are shown in fig.b
Note:
• Top of red + top of green = base of blue = 4
• So top of red = top of green = top of each = 4⁄2 = 2 units
Solved example 31.14
All the rectangles in fig.31.29 below have sides parallel to the axes.
 |
| Fig.31.29 |
Solution:
Fig.a:
1. To find the coordinates of B:
(i) Given that AB is parallel to the x-axis. So every point on AB will be at the same distance from the x-axis.
• So every point on AB will have the same y coordinate as A.
• The corner B is a point on AB. So B will have the same y coordinate as A.
• Thus we get: y coordinate of B is 3
(ii) Given that BC is parallel to the y-axis. So every point on BC will be at the same distance from the y-axis.
• So every point on BC will have the same x coordinate as A.
• The corner B is a point on BC. So B will have the same x coordinate as C.
• Thus we get: x coordinate of B is 2
(iii) From (i) and (ii) we get: The coordinates of B are: (2,3)
2. To find the coordinates of D:
(i) Given that CD is parallel to the x-axis. So every point on CD will be at the same distance from the x-axis.
• So every point on CD will have the same y coordinate as C.
• The corner D is a point on CD. So D will have the same y coordinate as C.
• Thus we get: y coordinate of D is 4
(ii) Given that AD is parallel to the y-axis. So every point on AD will be at the same distance from the y-axis.
• So every point on AD will have the same x coordinate as A.
• The corner D is a point on AD. So D will have the same x coordinate as A.
• Thus we get: x coordinate of D is -2
(iii) From (i) and (ii) we get: The coordinates of B are: (-2,4)
Fig.b:
1. To find the coordinates of A:
(i) Given that AB is parallel to the x-axis. So every point on AB will be at the same distance from the x-axis.
• So every point on AB will have the same y coordinate as B.
• The corner A is a point on AB. So A will have the same y coordinate as B.
• Thus we get: y coordinate of A is -4
(ii) Given that AD is parallel to the y-axis. So every point on AD will be at the same distance from the y-axis.
• So every point on AD will have the same x coordinate as D.
• The corner A is a point on AD. So A will have the same x coordinate as D.
• Thus we get: x coordinate of A is -1
(iii) From (i) and (ii) we get: The coordinates of B are: (-1,-4)
2. To find the coordinates of C:
(i) Given that CD is parallel to the x-axis. So every point on CD will be at the same distance from the x-axis.
• So every point on CD will have the same y coordinate as D.
• The corner C is a point on CD. So C will have the same y coordinate as D.
• Thus we get: y coordinate of C is -2
(ii) Given that BC is parallel to the y-axis. So every point on BC will be at the same distance from the y-axis.
• So every point on BC will have the same x coordinate as B.
• The corner C is a point on BC. So C will have the same x coordinate as B.
• Thus we get: x coordinate of C is 2
(iii) From (i) and (ii) we get: The coordinates of C are: (2,-2)
Fig.c:
1. To find the coordinates of B:
(i) Given that AB is parallel to the x-axis. So every point on AB will be at the same distance from the x-axis.
• So every point on AB will have the same y coordinate as A.
• The corner B is a point on AB. So B will have the same y coordinate as A.
• Thus we get: y coordinate of B is 3
(ii) Given that BC is parallel to the y-axis. So every point on BC will be at the same distance from the y-axis.
• So every point on BC will have the same x coordinate as A.
• The corner B is a point on BC. So B will have the same x coordinate as C.
• Thus we get: x coordinate of B is 2
(iii) From (i) and (ii) we get: The coordinates of B are: (2,3)
2. To find the coordinates of D:
(i) Given that CD is parallel to the x-axis. So every point on CD will be at the same distance from the x-axis.
• So every point on CD will have the same y coordinate as C.
• The corner D is a point on CD. So D will have the same y coordinate as C.
• Thus we get: y coordinate of D is 6
(ii) Given that AD is parallel to the y-axis. So every point on AD will be at the same distance from the y-axis.
• So every point on AD will have the same x coordinate as A.
• The corner D is a point on AD. So D will have the same x coordinate as A.
• Thus we get: x coordinate of D is -1
(iii) From (i) and (ii) we get: The coordinates of B are: (-1,6)
The completed fig.31.30 is shown below:
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| Fig.31.30 |
Given below are four pairs of coordinates. Each pair contains coordinates of any two corners of rectangles whose sides are parallel to the axes. In each pair, mark each coordinate as top-left, top-right, bottom-left or bottom-right. Mark them with out drawing axes. After marking, find the other coordinates also.
(i) (3,5), (7,8)
(ii) (6,2), (5,4)
(iii) (-3,5), (-7,1)
(iv) (-1,-2), (-5,-4)
Solution:
Case (i):
1. Consider the x coordinates: 3 and 7
(i) 7 is greater than 3. So (7,8) is on the right side of (3,5)
(ii) Thus the possibilities for (3,5) are: left top or left bottom
(iii) Also possibilities for (7,8) are: right top or right bottom
2. Consider the y coordinates: 5 and 8
(i) 8 is greater than 5. So (3,5) is at bottom and (7,8) is at top
(ii) Thus the possibilities for (3,5) are: left bottom or right bottom
(iii) Also possibilities for (7,8) are: left top or right top
3. Combining 1(ii) and 2(ii) we get: (3,5) is left bottom [Taking the common possibility. See the green highlighted lines]
4. Combining 1(iii) and 2(iii) we get: (7,8) is right top [Taking the common possibility. See the red highlighted lines]
This is shown in the fig.31.31 below:
 |
| Fig.31.31 |
1. Consider the x coordinates: 6 and 5
(i) 6 is greater than 5. So (6,2) is on the right side of (5,4)
(ii) Thus the possibilities for (5,4) are: left top or left bottom
(iii) Also possibilities for (6,2) are: right top or right bottom
2. Consider the y coordinates: 2 and 4
(i) 5 is greater than 2. So (6,2) is at bottom and (5,4) is at top
(ii) Thus the possibilities for (6,2) are: left bottom or right bottom
(iii) Also possibilities for (5,4) are: left top or right top
3. Combining 1(ii) and 2(iii) we get: (5,4) is left top [Taking the common possibility]
4. Combining 1(iii) and 2(ii) we get: (6,2) is right bottom [Taking the common possibility]
This is shown in the fig. above
Case (iii):
1. Consider the x coordinates: -3 and -7
(i) -3 is greater than -7. So (-3,5) is on the right side of (-7,1)
(ii) Thus the possibilities for (-7,1) are: left top or left bottom
(iii) Also possibilities for (-3,-5) are: right top or right bottom
2. Consider the y coordinates: 5 and 1
(i) 5 is greater than 1. So (-7,1) is at bottom and (-3,5) is at top
(ii) Thus the possibilities for (-7,1) are: left bottom or right bottom
(iii) Also possibilities for (-3,5) are: left top or right top
3. Combining 1(ii) and 2(ii) we get: (-7,1) is left bottom [Taking the common possibility]
4. Combining 1(iii) and 2(ii) we get: (-3,5) is right top [Taking the common possibility]
This is shown in the fig. above
Case (iv):
1. Consider the x coordinates: -1 and -5
(i) -1 is greater than -5. So (-1,-2) is on the right side of (-5,-4)
(ii) Thus the possibilities for (-5,4) are: left top or left bottom
(iii) Also possibilities for (-1,-2) are: right top or right bottom
2. Consider the y coordinates: -2 and -4
(i) -2 is greater than -4. So (-5,-4) is at bottom and (-1,-2) is at top
(ii) Thus the possibilities for (-5,-4) are: left bottom or right bottom
(iii) Also possibilities for (-1,-2) are: left top or right top
3. Combining 1(ii) and 2(ii) we get: (-5,-4) is left bottom [Taking the common possibility]
4. Combining 1(iii) and 2(ii) we get: (-1,-2) is right top [Taking the common possibility]
This is shown in the fig. above
Part 2:
The remaining coordinates of each rectangle can be calculated by using the procedure that we saw in the previous example. The completed figs are shown below:
 |
| Fig.31.32 |
In the next section we will see how to calculate the distance between any two points when their coordinates are given.
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