# Slope of a Line

The slope of a line determines how a line is slanting with respect to X-axis. Any line that is not parallel to X- axis must cross X-axis somewhere and make an angle with the X-axis.

The slope of a line is also known as its gradient and is a constant for that particular line.

**Remark:**

- Since X-axis makes angle with itself, the slope of X-axis is zero.
- Y-axis is perpendicular to X-axis; therefore the slope of Y-axis is not defined.

**Let be any two points on the given line making an angle with the X-axis. Draw PR and QS perpendicular to X-axis and PM âŠ¥ QS.**

# Slope of a line joining any two points on the line

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âˆ´ The slope of are tan Î¸ . We can conclude that if the slopes of two (or more) lines are equal, they are parallel and conversely if the lines are parallel, their slopes are equal.

If are the slopes of two parallel lines, then

# Slopes of parallel lines

Let be two parallel lines. Since lines are parallel, they should have the same inclination (say Î¸ )

âˆ´ The slope of are tan Î¸ . We can conclude that if the slopes of two (or more) lines are equal, they are parallel and conversely if the lines are parallel, their slopes are equal.

If are the slopes of two parallel lines, then

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âˆ´ Two lines of slopes are perpendicular if and conversely if , the lines with slopes are perpendicular. Hence two (non-vertical) lines are perpendicular to each other if and only if their slopes are negative reciprocals of each other.

# Slopes of perpendicular lines

Let be two perpendicular lines, making angles with the X-axis.âˆ´ Two lines of slopes are perpendicular if and conversely if , the lines with slopes are perpendicular. Hence two (non-vertical) lines are perpendicular to each other if and only if their slopes are negative reciprocals of each other.

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Let us consider two lines in a plane. They will intersect or will be parallel. We shall consider two intersecting lines.

Since we can find the acute angle (which will give the obtuse angle which is the supplement of the acute angle) we generally consider
If the lines are perpendicular,

Find the slope of the line with angle of inclination with X - axis

Find the slope of the line joining

Find whether the lines drawn through the two pairs of points are parallel or perpendicular.

(5, 2), (0, 5) and (0, 0), (âˆ’ 5, 3)

b) (âˆ’ 3, 1), (âˆ’ 2, âˆ’ 2) and (2, 5), (8, 7)

Using the idea of slopes, prove that the points

(4, 3), (2, 1) and (7, 6)

b) (3, âˆ’ 6), (2, âˆ’ 4) and (âˆ’ 4, 8) are collinear.

a) Let the points be A(4, 3), B(2, 1) and C(7, 6).

b) Let the three points be A(3, âˆ’ 6), B(2, âˆ’ 4) and C(âˆ’ 4, 8)

The vertices of a Î” ABC are A(âˆ’ 3, 3), B(âˆ’ 1, âˆ’ 4) and C(5, âˆ’ d2). Find (a) the slopes of the altitudes (b) the slopes of the medians (c) the slopes of the perpendicular bisectors.

a)

c) Perpendicular bisectors are also perpendicular to the bases. Hence the slope of each perpendicular bisector will be the same as that of the corresponding altitude. As you can see in the diagram, the blue lines are the perpendicular bisectors and they are parallel to the altitudes which are the red lines.

Show that triangle with vertices (0, âˆ’ 1), (âˆ’ 2, 3) and (6, 7) is right angled, without using Pythagorean theorem. Find the 4

Let the points be A(0, âˆ’ 1), B(âˆ’ 2, 3), C(6, 7)

Show that the points (8, 2), (5, âˆ’ 3) and (0, 0) form the vertices of a right angled isosceles triangle.

Let A(8, 2), B(5, âˆ’ 3) and C(0, 0) be the vertices

# Angle between two lines

Let us consider two lines in a plane. They will intersect or will be parallel. We shall consider two intersecting lines.

Let be the two lines, having inclinations . Let them meet at an acute angle and obtuse angle between them.

Since we can find the acute angle (which will give the obtuse angle which is the supplement of the acute angle) we generally consider

**Note:** This formula also enables you to find the condition for parallelism and perpendicularity of two lines. If the lines are parallel,

**Example:1**

Find the slope of the line with angle of inclination with X - axis

**Solution:**

**Example: 2**

Find the slope of the line joining

**Solution:**

**Example: 3**

Find whether the lines drawn through the two pairs of points are parallel or perpendicular.

(5, 2), (0, 5) and (0, 0), (âˆ’ 5, 3)

b) (âˆ’ 3, 1), (âˆ’ 2, âˆ’ 2) and (2, 5), (8, 7)

**Solution:**

**â€‹Example: 4**

Using the idea of slopes, prove that the points

(4, 3), (2, 1) and (7, 6)

b) (3, âˆ’ 6), (2, âˆ’ 4) and (âˆ’ 4, 8) are collinear.

**Solution:**

a) Let the points be A(4, 3), B(2, 1) and C(7, 6).

b) Let the three points be A(3, âˆ’ 6), B(2, âˆ’ 4) and C(âˆ’ 4, 8)

**Example: 5**

The vertices of a Î” ABC are A(âˆ’ 3, 3), B(âˆ’ 1, âˆ’ 4) and C(5, âˆ’ d2). Find (a) the slopes of the altitudes (b) the slopes of the medians (c) the slopes of the perpendicular bisectors.

**Solution:**

a)

b)

D, E, F are the mid-points of sides BC, CA and ABc) Perpendicular bisectors are also perpendicular to the bases. Hence the slope of each perpendicular bisector will be the same as that of the corresponding altitude. As you can see in the diagram, the blue lines are the perpendicular bisectors and they are parallel to the altitudes which are the red lines.

**Example: 6**

Show that triangle with vertices (0, âˆ’ 1), (âˆ’ 2, 3) and (6, 7) is right angled, without using Pythagorean theorem. Find the 4

^{th}vertex which will make a rectangle with the given vertices.

**Solution:**

Let the points be A(0, âˆ’ 1), B(âˆ’ 2, 3), C(6, 7)

**Example:7**

Show that the points (8, 2), (5, âˆ’ 3) and (0, 0) form the vertices of a right angled isosceles triangle.

**Solution:**

Let A(8, 2), B(5, âˆ’ 3) and C(0, 0) be the vertices

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