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PROPERTIES OF LINES EXPLAINED

A straight line can be defined as the curve which is such that the line segment joining any two points on it lies wholly on it.


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DEFINITION

A straight line can be defined as the curve which is such that the line segment joining any two points on it lies wholly on it.

ANGLE OF INCLINATION OF A LINE

The angle of inclination of a line is the measure of the angle between the x-axis and the line measured in the anticlockwise direction.

When two lines are parallel, they have the same inclination.

The inclination of a line which is parallel to the x-axis or coinciding with the x-axis is 0º

The angle of inclination of the line lies between 0º and 180º i.e., 0<??? and ?? ?/2.

SLOPE OR GRADIENT OF A LINE

If inclination of a line is (??90º) then tan? is called the slope or gradient of the line, it is usually denoted by m.

? is positive or negative according as it is measured in anticlockwise or clockwise direction.

i.e., Slope of AB = m of AB = m(AB)

= tan ? or tan (- {?-?)}

= tan (?+?)

= slope of BA = m of BA

= m(BA)

m(AB) = m(BA)

Hence we do not take into consideration the direction of a line segment while talking on it’s slope.

Slope of a line is not the angle but is the tangent of the inclination of the line.

If a line is parallel to x-axis then its slope = tan0º = 0.

Slope of a line parallel to y-axis or perpendicular to x-axis is not defined.

If a line is equally inclined to the axes, then it will make the angle of 45º or 135º with the positive direction of x-axis. Slope in this case will be tan 45º or tan 135º i.e., +1 or -1. (1 + m1.m2).

ANGLE BETWEEN TWO LINES

 

Note:

1.) If two lines, whose slopes are m1 and m2 are parallel, iff

? =0o ? tan ? = 0

?

Thus when two lines are parallel, their slopes are equal

2.) If two lines whose slopes are m1 and m2 are perpendicular, iff

? = ?/2 ( or –?/2) ? cot? = 0 ?  m1.m= -1.

Thus, when two lines are perpendicular, the product of their slopes is -1. The slopes of each is the negative reciprocal of the slope  of the other, i.e., if m is the slope of a line, then the slope of a line perpendicular to it is -1/m.