Slope of a Straight Line

Slope (or gradient) of a straight line

If ( 90°) is the inclination of a straight line, then tan is called its slope (or gradient). The slope of a line is usually denoted by m.

Remark. Since tan is not defined when = 90°, therefore, the slope of a vertical line is not defined.

Slope of the line joining two points

The slope m of a non-vertical line passing through the points P(x1 , y1) and Q(x1, y1) is given by
     slope = m = (y2 -y1)/(x2 -x1)

Remarks

Illustrative Examples

Example

Without using Pythagoras theorem, show that the points A (1, 2), B (4, 5) and C (6, 3) are the vertices of a right-angled triangle.

Solution

In ABC, we have
m1 = slope of AB = (5 -2)/(4-1) = 3/3 = 1 and
m2 = slope of BC = (3 -5)/(6-4) = 2/2 = -1
m1 m2 = 1. (-1) = -1    => ABBC
Hence, the given points are the vertices of a right-angled triangle.

Example

Using slopes, show that the points A (6, -1), B (5, 0) and C (2, 3) are collinear.

Solution

Slope of AB = [0 -(-1)]/(5-6) = 1/(-1) = -1
and slope of AC = [3 -(-1)]/(2-6) = 4/(-4) = -1
=>  slope of AB = slope of AC
=>  AB and AC are parallel.
But AB and AC have point A is common, therefore, the given points A, B and C are collinear.

Exercise

  1. Find the slope of a line whose inclination is:
    (i) 30°         (ii) 2 /3     (iii) /3
  2. Find the inclination of a line whose gradient is:
    (i) 1/ 3     (ii) -1       (iii) - 3
  3. Find the gradient of the line containing the points
    (i) (-2, 3) and (5, -7)             (ii) (3, -7) and (0, 2)
  4. A line passes through the points (4, -6) and (-2, -5). Does it make an acute angle with the positive direction of x-axis?
  5. Find the equation of the locus of all points P such that the slope of the line joining origin and P is - 2.
  6. Show that the line joining (2, -3) and (-5, 1) is
    (i) parallel to the line joining (7, -1) and (0, 3)
    (ii) perpendicular to the line joining (4, 5) and (0, - 2)
  7. State, whether the two lines in each of the following problems are parallel, perpendicular or neither:
    (i) through (2, -5) and (-2, 5); through (6, 3) and (1, 1)
    (ii) through (5, 6) and (2, 3); through (9, -2) and (6, - 5)
    (iii) through (8, 2) and (-5, 3); through (16, 6) and (3, 15)
  8. Find y if the slope of the line joining (-8, 11), (2, y) is - 4/3.
  9. Find the value of x so that the line through (x, 9) and (2, 7) is parallel to the line through (2, - 2) and (6, 4).
  10. Without using Pythagoras theorem, show that the points (4, 4), (3, 5) and (-1, -1) are the vertices of a right angled triangle.

Answers

1. (i) 1/ 3            (ii) - 3        (iii) not defined
2. (i) 30°                (ii) 135°        (iii) 120°
3. (i)- 10/7              (ii) -3
4. No                      5. 2x +y = 0
7. (i) Perpendicular  (ii) parallel        (iii) neither
8. -7/3                    9. 10./3