Page 131 - DMTH201_Basic Mathematics-1

Basic Mathematics – I

Notes 3. In particular, if m = n, the coordinates of the mid-point of the line segment joining the
x x y y
points (x , y ) and (x , y ) are 1 2 , 1 2 .
1 1 2 2
2 2
4. Area of the triangle whose vertices are (x y ), (x , y ) and (x , y ) is
1, 1 2 2 3 3
1
x y y x y y x y y 2 .
2
1
1
1
3
2
3
3
2
For example, the area of the triangle, whose vertices are (4, 4), (3, – 2) and (– 3, 16) is
1 54
4( 2 16) 3(16 4) ( 3)(4 2) 27.
2 2
Notes If the area of the triangle ABC is zero, then three points A, B and C lie on a line, i.e.,
they are collinear.
In the this unit, we shall continue the study of coordinate geometry to study properties of
the simplest geometric figure – straight line. Despite its simplicity, the line is a vital concept
of geometry and enters into our daily experiences in numerous interesting and useful
ways. Main focus is on representing the line algebraically, for which slope is most essential.
5.2 Slope of a Line

As you are already familiar with coordinate geometry. A line in a coordinate plane forms two
angles with the x-axis, which are supplementary. The slope of a line is a number that measures
its “steepness”, usually denoted by the letter m. It is the change in y for a unit change in x along
the line. The angle (say) made by the line with positive direction of x-axis and measured anti-
clock-wise is called the inclination of the line. Obviously 0° 180° (Figure 5.2). If a line passes
through two distinct points P (x , y ) and P (x , y ), its slope is given by: m = (y – y ) / (x – x )
1 1 1 2 2 2 2 1 2 1
with x not equal to x
2 1
We observe that lines parallel to x-axis, or coinciding with x-axis, have inclination of 0°. The
inclination of a vertical line (parallel to or coinciding with y-axis) is 90°.

Figure 5.2

Definition 1: If is the inclination of a line l, then tan is called the slope or gradient of the line l.
The slope of a line whose inclination is 90° is not defined.
The slope of a line is denoted by m.

Thus, m = tan , 90°
It may be observed that the slope of x-axis is zero and slope of y-axis is not defined.

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