Summary

Videos

References

In a coordinate plane with two points A (x_{1}, y_{1}) and B (x_{2}, y_{2}),

Distance formula: AB = âˆš(*x*_{2} - *x*_{1})^{2} + (*y*_{2} - *y*_{1})^{2}

Ratio formula: Coordinates of a point C dividing line segment AB internally in the ratio m:n = [mx_{2} + nx_{1}/m + n , my_{2} + ny_{1}/m + n].

If m = n, coordinates of C = [x_{1} + x_{2}/2, y_{1} + y_{2}/2].

Area of triangle: Area of âˆ†ABC = Â½ âˆ£*x*_{1}(*y*_{2} - *y*_{3}) + *x*_{2}(*y*_{3} - *y*_{1}) + *x*_{3}(*y*_{1} - *y*_{2})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 90Â°. The inclination of a line can have a value anywhere from zero to 180Â°.

Slope of line AB (*m*) = tan Î¸

If Î¸ = 0^{o}

*m* = tan 0^{o} = 0

If Î¸ = 90^{o}

*m* = tan 90^{o} â‡’ Not defined.

Slope of a line passing through two given points:

Let P (x_{1}, y_{1}) and Q (x_{2}, y_{2}) be two points on a non-vertical line l, whose inclination is Î¸.

Since the line is not a vertical line, x_{1} â‰ x_{2.}

Case I: 0^{o} â‰¤ Î¸ < 90^{o}

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 = y_{2} - y_{1} â€¦â€¦â€¦â€¦â€¦â€¦â€¦ (2)

PM = TR

PM = OR - OT

PM = x_{2} - x_{1} â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦ (3)

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

Tan Î¸ = (y_{2} - y_{1})/(x_{2} - x_{1}).

Case II: 90^{o} < Î¸ â‰¤ 180^{o}

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 = 180^{o } (Supplementary angles)

âˆ´ âˆ QPM = 180^{o} 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 = (y_{2} - y_{1})/(x_{2} - x_{1}) â€¦.(2)

From equations 1 and 2:

m = -(y_{2} - y_{1})/(x_{2} - x_{1})

= (y_{2} - y_{1})/(x_{2} - x_{1})

m = (y_{2} - y_{1})/(x_{2} - x_{1})

Conditions for parallelism and perpendicularity

Consider two non-vertical parallel lines l_{1} and l_{2} having inclination a and b, respectively.

If line l_{1} and l_{2} are parallel, their inclination must be the same.

â‡’ Î± = Î²

â‡’ tan Î± = tan Î²

tan Î± = *m*_{1} and tan Î² = *m*_{2}

â‡’ *m*_{1} = *m*_{2}

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 *m*_{1} = *m*_{2 }â‡’ *l*_{1 }|| *l*_{2}

Relationship between the slopes of perpendicular lines:

Given: *l*_{1} âŠ¥ *l*_{2}

â‡’ Î² = Î± + 90^{o}

â‡’ tan Î² = tan (Î± + 90^{o})

â‡’ tan Î² = - cot Î±

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

tan Î± = *m*_{1} and tan Î² = *m*_{2}

â‡’ *m*_{2} = -1/*m*_{1}

â‡’ *m*_{1} *m*_{2} = -1

Given: *m*_{1} *m*_{2} = -1

â‡’ *l*_{1} âŠ¥ *l*_{2}

In a coordinate plane with two points A (x_{1}, y_{1}) and B (x_{2}, y_{2}),

Distance formula: AB = âˆš(*x*_{2} - *x*_{1})^{2} + (*y*_{2} - *y*_{1})^{2}

Ratio formula: Coordinates of a point C dividing line segment AB internally in the ratio m:n = [mx_{2} + nx_{1}/m + n , my_{2} + ny_{1}/m + n].

If m = n, coordinates of C = [x_{1} + x_{2}/2, y_{1} + y_{2}/2].

Area of triangle: Area of âˆ†ABC = Â½ âˆ£*x*_{1}(*y*_{2} - *y*_{3}) + *x*_{2}(*y*_{3} - *y*_{1}) + *x*_{3}(*y*_{1} - *y*_{2})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 90Â°. The inclination of a line can have a value anywhere from zero to 180Â°.

Slope of line AB (*m*) = tan Î¸

If Î¸ = 0^{o}

*m* = tan 0^{o} = 0

If Î¸ = 90^{o}

*m* = tan 90^{o} â‡’ Not defined.

Slope of a line passing through two given points:

Let P (x_{1}, y_{1}) and Q (x_{2}, y_{2}) be two points on a non-vertical line l, whose inclination is Î¸.

Since the line is not a vertical line, x_{1} â‰ x_{2.}

Case I: 0^{o} â‰¤ Î¸ < 90^{o}

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 = y_{2} - y_{1} â€¦â€¦â€¦â€¦â€¦â€¦â€¦ (2)

PM = TR

PM = OR - OT

PM = x_{2} - x_{1} â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦ (3)

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

Tan Î¸ = (y_{2} - y_{1})/(x_{2} - x_{1}).

Case II: 90^{o} < Î¸ â‰¤ 180^{o}

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 = 180^{o } (Supplementary angles)

âˆ´ âˆ QPM = 180^{o} 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 = (y_{2} - y_{1})/(x_{2} - x_{1}) â€¦.(2)

From equations 1 and 2:

m = -(y_{2} - y_{1})/(x_{2} - x_{1})

= (y_{2} - y_{1})/(x_{2} - x_{1})

m = (y_{2} - y_{1})/(x_{2} - x_{1})

Conditions for parallelism and perpendicularity

Consider two non-vertical parallel lines l_{1} and l_{2} having inclination a and b, respectively.

If line l_{1} and l_{2} are parallel, their inclination must be the same.

â‡’ Î± = Î²

â‡’ tan Î± = tan Î²

tan Î± = *m*_{1} and tan Î² = *m*_{2}

â‡’ *m*_{1} = *m*_{2}

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 *m*_{1} = *m*_{2 }â‡’ *l*_{1 }|| *l*_{2}

Relationship between the slopes of perpendicular lines:

Given: *l*_{1} âŠ¥ *l*_{2}

â‡’ Î² = Î± + 90^{o}

â‡’ tan Î² = tan (Î± + 90^{o})

â‡’ tan Î² = - cot Î±

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

tan Î± = *m*_{1} and tan Î² = *m*_{2}

â‡’ *m*_{2} = -1/*m*_{1}

â‡’ *m*_{1} *m*_{2} = -1

Given: *m*_{1} *m*_{2} = -1

â‡’ *l*_{1} âŠ¥ *l*_{2}