In the previous section we completed the discussion on section formula. We also saw some solved examples. In this section we will see details about lines.
1. Consider any one point P on a plane.
• A line is to be drawn on that plane.
• That line should satisfy one condition.
♦ The condition is: The line should pass through P
• It is easy to drawn the line satisfying the condition.
• All we need to do is: Draw the line through P. This is shown in fig.34.18(a) below:
• In fact, we can draw infinite number of lines through that point P.
♦ All those lines will satisfy the given condition. This is shown in fig.34.18(b) above
2. Now consider any two given points A and B in a plane.
• A line is to be drawn on that plane.
• That line should satisfy one condition.
♦ The condition is: The line should pass through both A and B
• It is easy to drawn a line satisfying the condition.
• All we need to do is: Draw a line through A and B
• But this time we can drawn only one line satisfying the condition.
♦ No other line will pass through both A and B
• That line will be unique. Because, it will have a particular direction.
• The 'particular direction' may be any one of the following three cases:
♦ It may be vertical (see fig.34.19.a below)
♦ It may be horizontal (see fig.34.19.b below)
♦ It may be inclined (see fig.34.19.c below)
Let us now analyse each of the three cases:
Case 1:
1. In fig.34.19(a) below, a line passes through two points A and B.
• This line is shown in magenta color.
• It is a vertical line
• It satisfies one condition: The line must pass through both A and B
2. Can any other line satisfy the condition? Let us check:
Two other vertical lines are shown.
• One in cyan color and the other in red color.
• None of them will ever pass through A and B.
3. Apart from line AB, we can draw an infinite number of vertical lines.
• None of them will ever pass through both A and B
4. In the same way consider horizontal lines.
• There are infinite number of horizontal lines possible.
• Some of them may pass through either A OR B.
• But none of the horizontal lines will ever pass through both A and B
5. Now consider inclined lines. There are infinite number of inclined lines possible.
• Some of them may pass through either A OR B.
• But none of the inclined lines will ever pass through both A and B
■ So the line AB is unique. No other line will satisfy the condition
■ Note that all points on a vertical line (or line parallel to the y axis) will have the same x coordinate. But y coordinates will be different
Case 2:
1. In fig.34.19(b), a line passes through two points C and D.
• This line is shown in magenta color.
• It is a horizontal line
• It satisfies one condition: The line must pass through both C and D
2. Can any other line satisfy the condition? Let us check:
Two other horizontal lines are shown.
• One in cyan color and the other in red color.
• None of them will ever pass through C and D.
3. Apart from line CD, we can draw an infinite number of horizontal lines.
• But none of them will ever pass through both C and D
4. In the same way consider vertical lines.
• There are infinite number of vertical lines possible.
• Some of them may pass through either C OR D.
• But none of the vertical lines will ever pass through both C and D
5. Now consider inclined lines. There are infinite number of inclined lines possible.
• Some of them may pass through either C OR D.
• But none of the inclined lines will ever pass through both C and D
■ So the line CD is unique. No other line will satisfy the condition
■ Note that all points on a horizontal line (or line parallel to the x axis) will have the same y coordinate. But x coordinates will be different
Case 3:
1. In fig.34.19(c), a line passes through two points E and F.
• This line is shown in magenta color.
• It is an inclined line. It's angle of inclination with the horizontal is given as θ degrees
• It satisfies one condition: The line must pass through both E and F
2. Can any other line satisfy the condition? Let us check:
Two other inclined lines are shown.
• One in cyan color and the other in red color.
• They are parallel to the magenta line. That is., they make the same angle θ with the horizontal.
• But none of them will ever pass through E and F.
3. Apart from line EF, we can draw an infinite number of inclined lines parallel to EF itself.
• But none of them will ever pass through both E and F
4. But there can be infinite number of inclined lines which are not parallel to EF. Can any one of them satisfy the condition? Let us check:
• The angle θ can take any values like 26, 35, 42 etc.,
• But once the line pass through both E and F, the θ will get fixed.
• If after wards we change θ even by the smallest value, the line will not pass through both E and F.
♦ That is., the line will no longer satisfy the condition.
5. Now, we need not even consider about vertical or horizontal lines. They will never pass through both E and F
■ So the line EF is unique. No other line will satisfy the condition
■ Note that in an inclined line, both x and y coordinates will be different for different points on it.
Now we will learn about inclined lines in a little more detail:
1. Consider the inclined line in fig.34.20 below:
• Four random points A, B, C and D are marked on it
• The coordinates of A, B, C and D are (x1,y1), (x2,y2), (x3,y3) and (x4,y4) respectively
2. In fig.b, green lines are drawn.
• These green lines have a special property: They are all parallel to the axes
♦ The horizontal green lines are parallel to the x axis
♦ The vertical green lines are parallel to the y axis
3. The vertical and horizontal green lines at bottom intersect at B'.
• The vertical and horizontal green lines at top intersect at D'.
• We know that the angle between the axes will always be 90o
• Since the green lines are parallel to the axes, the angle at B' and D' will also be 90o
• So triangles AB'B and CD'D are right triangles
4. Now we will see the relation between the two right triangles:
• Let ∠BAB' be α. Then ∠DCD' will also be equal to α.
5. To establish the relation between the two triangles, we can use any one of the two methods:
• Applying the principles of Similar triangles
• Applying the principles of trigonometry
• In this discussion we will use the principles of trigonometry. However, readers are advised to write the steps using 'principles of triangles' in his/her own note books
6. We have already seen the basics of trigonometry here.
• In fig.34.20(c) above, consider the right triangle AB'B. Taking the tan ratio, we will get:
tan α = opposite side⁄adjacent side = BB'⁄AB'
7. Again in fig.34.20(c) above, consider the right triangle CD'D. Taking the tan ratio, we will get:
tan α = opposite side⁄adjacent side = DD'⁄CD'
8. Now, the tan in ⊿AB'B can be equated to the tan in ⊿CD'D. Because, both are taken for the same angle α
9. So we can write: BB'⁄AB' = DD'⁄CD'
10. Let us analyse the above result:
• BB' is the vertical travel from A to B
• AB' is the horizontal travel from A to B
♦ So BB'⁄AB' is the ratio: vertical travel from A to B⁄horizontal travel from A to B.
Similarly,
• DD' is the vertical travel from C to D
• CD' is the horizontal travel from C to D
♦ So DD'⁄CD' is the ratio: vertical travel from C to D⁄horizontal travel from C to D.
11. From (9) we have seen that the two ratios are equal. So we can write:
vertical travel from A to B⁄horizontal travel from A to B = vertical travel from C to D⁄horizontal travel from C to D.
■ This is a very important result. It can be written in a general form:
(i) Mark any number of points as we like on an inclined line. Like A, B, C, D, . . .
(ii) Group them into pairs. We can make pairs in any form. Order is not important.
For example: [A,B], [C,D], [A,C], [B,P], . . .
(iii) Take out any one pair
• Calculate the vertical travel from the first point in that pair to the second point
• Calculate the horizontal travel from the first point in that pair to the second point
(iv) Take the ratio: vertical travel⁄horizontal travel
• Let us call this ratio as 'm'. So we can write: m = vertical travel⁄horizontal travel.
(v) Calculate 'm' for each pair. We will find that 'm' is the same for all pairs.
An example is shown in fig.34.21 below:
• Take [A,B]:
m = vertical travel⁄horizontal travel = (5-2.75)⁄(4-1) = 2.25⁄3 = 0.75
• Take [B,C]:
m = vertical travel⁄horizontal travel = (6.5-5)⁄(6-4) = 1.5⁄2 = 0.75
• Take [C,E]:
m = vertical travel⁄horizontal travel = (9.5-6.5)⁄(10-6) = 3⁄4 = 0.75
• Take [B,D]:
m = vertical travel⁄horizontal travel = (8-5)⁄(8-4) = 3⁄4 = 0.75
(vi) So 'm' is constant for a line.
• In other words, every line will have a unique value for 'm'
■ It is called the slope of that line
(vii) So we can write:
■ Slope of a line = m = vertical travel from any point A to any other point B⁄horizontal travel from A to B
• If the coordinates of A are (x1,y1) and those of B are (x2,y2), then:
♦ the numerator = vertical travel = BB' = (y2-y1) [see fig.34.22 below]
♦ the denominator is = horizontal travel = AB' = (x2-x1)
■ So we can write:
Slope of a line = m = (y2-y1)⁄(x2-x1)
• In the above result, the numerator is 'opposite side in the triangle'
• The denominator is 'adjacent side in the triangle'
• But = opposite side⁄adjacent side = tan α
• So Slope can also be obtained as:
■ Slope of a line = m = tan α.
• Where α is the angle which the line makes with the horizontal
12. So now we know how to calculate the slope of a line.
• All we need is the coordinates of any two points on that line.
OR, the angle which the line makes with the horizontal
In the next section, we will see how the slope 'm' can be put to practical use.
1. Consider any one point P on a plane.
• A line is to be drawn on that plane.
• That line should satisfy one condition.
♦ The condition is: The line should pass through P
• It is easy to drawn the line satisfying the condition.
• All we need to do is: Draw the line through P. This is shown in fig.34.18(a) below:
 |
| Fig.34.18 |
♦ All those lines will satisfy the given condition. This is shown in fig.34.18(b) above
2. Now consider any two given points A and B in a plane.
• A line is to be drawn on that plane.
• That line should satisfy one condition.
♦ The condition is: The line should pass through both A and B
• It is easy to drawn a line satisfying the condition.
• All we need to do is: Draw a line through A and B
• But this time we can drawn only one line satisfying the condition.
♦ No other line will pass through both A and B
• That line will be unique. Because, it will have a particular direction.
• The 'particular direction' may be any one of the following three cases:
♦ It may be vertical (see fig.34.19.a below)
♦ It may be horizontal (see fig.34.19.b below)
♦ It may be inclined (see fig.34.19.c below)
Let us now analyse each of the three cases:
Case 1:
1. In fig.34.19(a) below, a line passes through two points A and B.
• This line is shown in magenta color.
• It is a vertical line
 |
| Fig.34.19 |
2. Can any other line satisfy the condition? Let us check:
Two other vertical lines are shown.
• One in cyan color and the other in red color.
• None of them will ever pass through A and B.
3. Apart from line AB, we can draw an infinite number of vertical lines.
• None of them will ever pass through both A and B
4. In the same way consider horizontal lines.
• There are infinite number of horizontal lines possible.
• Some of them may pass through either A OR B.
• But none of the horizontal lines will ever pass through both A and B
5. Now consider inclined lines. There are infinite number of inclined lines possible.
• Some of them may pass through either A OR B.
• But none of the inclined lines will ever pass through both A and B
■ So the line AB is unique. No other line will satisfy the condition
■ Note that all points on a vertical line (or line parallel to the y axis) will have the same x coordinate. But y coordinates will be different
Case 2:
1. In fig.34.19(b), a line passes through two points C and D.
• This line is shown in magenta color.
• It is a horizontal line
• It satisfies one condition: The line must pass through both C and D
2. Can any other line satisfy the condition? Let us check:
Two other horizontal lines are shown.
• One in cyan color and the other in red color.
• None of them will ever pass through C and D.
3. Apart from line CD, we can draw an infinite number of horizontal lines.
• But none of them will ever pass through both C and D
4. In the same way consider vertical lines.
• There are infinite number of vertical lines possible.
• Some of them may pass through either C OR D.
• But none of the vertical lines will ever pass through both C and D
5. Now consider inclined lines. There are infinite number of inclined lines possible.
• Some of them may pass through either C OR D.
• But none of the inclined lines will ever pass through both C and D
■ So the line CD is unique. No other line will satisfy the condition
■ Note that all points on a horizontal line (or line parallel to the x axis) will have the same y coordinate. But x coordinates will be different
Case 3:
1. In fig.34.19(c), a line passes through two points E and F.
• This line is shown in magenta color.
• It is an inclined line. It's angle of inclination with the horizontal is given as θ degrees
• It satisfies one condition: The line must pass through both E and F
2. Can any other line satisfy the condition? Let us check:
Two other inclined lines are shown.
• One in cyan color and the other in red color.
• They are parallel to the magenta line. That is., they make the same angle θ with the horizontal.
• But none of them will ever pass through E and F.
3. Apart from line EF, we can draw an infinite number of inclined lines parallel to EF itself.
• But none of them will ever pass through both E and F
4. But there can be infinite number of inclined lines which are not parallel to EF. Can any one of them satisfy the condition? Let us check:
• The angle θ can take any values like 26, 35, 42 etc.,
• But once the line pass through both E and F, the θ will get fixed.
• If after wards we change θ even by the smallest value, the line will not pass through both E and F.
♦ That is., the line will no longer satisfy the condition.
5. Now, we need not even consider about vertical or horizontal lines. They will never pass through both E and F
■ So the line EF is unique. No other line will satisfy the condition
■ Note that in an inclined line, both x and y coordinates will be different for different points on it.
Now we will learn about inclined lines in a little more detail:
1. Consider the inclined line in fig.34.20 below:
• Four random points A, B, C and D are marked on it
 |
| Fig.34.20 |
2. In fig.b, green lines are drawn.
• These green lines have a special property: They are all parallel to the axes
♦ The horizontal green lines are parallel to the x axis
♦ The vertical green lines are parallel to the y axis
3. The vertical and horizontal green lines at bottom intersect at B'.
• The vertical and horizontal green lines at top intersect at D'.
• We know that the angle between the axes will always be 90o
• Since the green lines are parallel to the axes, the angle at B' and D' will also be 90o
• So triangles AB'B and CD'D are right triangles
4. Now we will see the relation between the two right triangles:
• Let ∠BAB' be α. Then ∠DCD' will also be equal to α.
5. To establish the relation between the two triangles, we can use any one of the two methods:
• Applying the principles of Similar triangles
• Applying the principles of trigonometry
• In this discussion we will use the principles of trigonometry. However, readers are advised to write the steps using 'principles of triangles' in his/her own note books
6. We have already seen the basics of trigonometry here.
• In fig.34.20(c) above, consider the right triangle AB'B. Taking the tan ratio, we will get:
tan α = opposite side⁄adjacent side = BB'⁄AB'
7. Again in fig.34.20(c) above, consider the right triangle CD'D. Taking the tan ratio, we will get:
tan α = opposite side⁄adjacent side = DD'⁄CD'
8. Now, the tan in ⊿AB'B can be equated to the tan in ⊿CD'D. Because, both are taken for the same angle α
9. So we can write: BB'⁄AB' = DD'⁄CD'
10. Let us analyse the above result:
• BB' is the vertical travel from A to B
• AB' is the horizontal travel from A to B
♦ So BB'⁄AB' is the ratio: vertical travel from A to B⁄horizontal travel from A to B.
Similarly,
• DD' is the vertical travel from C to D
• CD' is the horizontal travel from C to D
♦ So DD'⁄CD' is the ratio: vertical travel from C to D⁄horizontal travel from C to D.
11. From (9) we have seen that the two ratios are equal. So we can write:
vertical travel from A to B⁄horizontal travel from A to B = vertical travel from C to D⁄horizontal travel from C to D.
■ This is a very important result. It can be written in a general form:
(i) Mark any number of points as we like on an inclined line. Like A, B, C, D, . . .
(ii) Group them into pairs. We can make pairs in any form. Order is not important.
For example: [A,B], [C,D], [A,C], [B,P], . . .
(iii) Take out any one pair
• Calculate the vertical travel from the first point in that pair to the second point
• Calculate the horizontal travel from the first point in that pair to the second point
(iv) Take the ratio: vertical travel⁄horizontal travel
• Let us call this ratio as 'm'. So we can write: m = vertical travel⁄horizontal travel.
(v) Calculate 'm' for each pair. We will find that 'm' is the same for all pairs.
An example is shown in fig.34.21 below:
 |
| Fig.34.21 |
m = vertical travel⁄horizontal travel = (5-2.75)⁄(4-1) = 2.25⁄3 = 0.75
• Take [B,C]:
m = vertical travel⁄horizontal travel = (6.5-5)⁄(6-4) = 1.5⁄2 = 0.75
• Take [C,E]:
m = vertical travel⁄horizontal travel = (9.5-6.5)⁄(10-6) = 3⁄4 = 0.75
• Take [B,D]:
m = vertical travel⁄horizontal travel = (8-5)⁄(8-4) = 3⁄4 = 0.75
(vi) So 'm' is constant for a line.
• In other words, every line will have a unique value for 'm'
■ It is called the slope of that line
(vii) So we can write:
■ Slope of a line = m = vertical travel from any point A to any other point B⁄horizontal travel from A to B
• If the coordinates of A are (x1,y1) and those of B are (x2,y2), then:
♦ the numerator = vertical travel = BB' = (y2-y1) [see fig.34.22 below]
♦ the denominator is = horizontal travel = AB' = (x2-x1)
 |
| Fig.34.22 |
Slope of a line = m = (y2-y1)⁄(x2-x1)
• In the above result, the numerator is 'opposite side in the triangle'
• The denominator is 'adjacent side in the triangle'
• But = opposite side⁄adjacent side = tan α
• So Slope can also be obtained as:
■ Slope of a line = m = tan α.
• Where α is the angle which the line makes with the horizontal
12. So now we know how to calculate the slope of a line.
• All we need is the coordinates of any two points on that line.
OR, the angle which the line makes with the horizontal
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