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# Slope of a Line

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#### Slope of a Line - Lesson Summary

In a coordinate plane with two points A (x1, y1) and B (x2, y2),

Distance formula: AB = âˆš(x2 - x1)2 + (y2 - y1)2

Ratio formula: Coordinates of a point C dividing line segment AB internally in the ratio m:n = [mx2 + nx1/m + n , my2 + ny1/m + n].

If m = n, coordinates of C = [x1 + x2/2, y1 + y2/2].

Area of triangle: Area of âˆ†ABC = ½ âˆ£x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)âˆ£

If area of âˆ†ABC = 0 â‡’ A, B and C are collinear points.

A line is said to be inclined when it makes an angle with the horizontal. A line intersecting the X-axis forms supplementary angles.

The angle made by a straight line in the anti-clockwise direction with the X-axis is called inclination.

The anti-clockwise direction is also called the positive direction. The X- axis and the lines parallel to it are called horizontal lines

If line AB lies along the X-axis or is parallel to the X-axis, then its inclination is zero. The Y-axis and the lines parallel to it are called vertical lines.

If line AB is parallel to the Y-axis or perpendicular to the X-axis, then its inclination is 90°. The inclination of a line can have a value anywhere from zero to 180°.

Slope of line AB (m) = tan Î¸

If Î¸ = 0o

m = tan 0o = 0

If Î¸ = 90o

m = tan 90o â‡’ Not defined.

Slope of a line passing through two given points:

Let P (x1, y1) and Q (x2, y2) be two points on a non-vertical line l, whose inclination is Î¸.

Since the line is not a vertical line, x1 â‰  x2.

Case I: 0o â‰¤ Î¸ < 90o

Draw perpendiculars 'QR' and 'PT' from 'Q' and 'P' on to the X-axis.

Draw perpendicular 'PM' from 'P' on to 'QR'.

Since QR âŠ¥ PM and QR âŠ¥ X-axis,

PM || X-axis

â‡’ âˆ QSX = âˆ QPM = Î¸ (Corresponding angles)

In âˆ†PMQ:

tan Î¸ = QM/PM â€¦â€¦â€¦â€¦â€¦â€¦â€¦ (1)

We have

QM = QR - MR

QM = y2 - y1 â€¦â€¦â€¦â€¦â€¦â€¦â€¦ (2)

PM = TR

PM = OR - OT

PM = x2 - x1 â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦ (3)

Substituting the values of QM and PM from (2) and (3) in (1):

Tan Î¸ = (y2 - y1)/(x2 - x1).

Case II: 90o < Î¸ â‰¤ 180o

Draw perpendiculars 'QR' and 'PT' onto X-axis.

Draw perpendicular 'PM' from 'P' to 'QR' and extend 'MP' to 'S'.

Since QR âŠ¥ PM and QR âŠ¥ X-axis,

PM || X-axis

â‡’ âˆ QUX = âˆ QPS = Î¸ (Corresponding angles)

âˆ QPS + âˆ QPM = 180o (Supplementary angles)

âˆ´ âˆ QPM = 180o â€“ Î¸

â‡’ Î¸ = 180 â€“ âˆ QPM

Thus, m = tan Î¸= tan(180 - âˆ QPM)

m = â€“ tan âˆ QPM â€¦.(1)

In âˆ†PMQ:

tan âˆ QPM = QM/PM = QM/RT

â‡’ tan âˆ QPM = (y2 - y1)/(x2 - x1) â€¦.(2)

From equations 1 and 2:

m = -(y2 - y1)/(x2 - x1)

= (y2 - y1)/(x2 - x1)

m = (y2 - y1)/(x2 - x1)

Conditions for parallelism and perpendicularity

Consider two non-vertical parallel lines l1 and l2 having inclination a and b, respectively.

If line l1 and l2 are parallel, their inclination must be the same.

â‡’ Î± = Î²

â‡’ tan Î± = tan Î²

tan Î± = m1 and tan Î² = m2

â‡’ m1 = m2

Thus, we can say that if two non-vertical lines are parallel, their slopes are equal.

The converse is also true. That is, if the slopes of two non-vertical lines are equal, the two lines are parallel.

If m1 = mâ‡’ l1 || l2

Relationship between the slopes of perpendicular lines:

Given: l1 âŠ¥ l2

â‡’ Î² = Î± + 90o

â‡’ tan Î² = tan (Î± + 90o)

â‡’ tan Î² = - cot Î±

â‡’ tan Î² = - 1/tan Î± â€¦..Equation (1)

tan Î± = m1 and tan Î² = m2

â‡’ m2 = -1/m1

â‡’ m1 m2 = -1

Given: m1 m2 = -1

â‡’ l1 âŠ¥ l2