Slope, Angle of Inclination and Angle Between Two Lines - Analytic Geometry

* Slope of a Line

Formula:

 
(Slope Intercept Form)

y = mx + b

m = rise/run = (y_{2 -} y1) / (x_{2 -} x1)

Where:

x1 and y1 = coordinates of first point

x2 and y2 = coordinates of second point

Note:
- In solving for the slope of a line, there must be first and second (points needed) since there are uncountable points that forms a line. Terms such as "first" and "second" points are just used to avoid confusion because we have to be consistent on what values to be substituted.

Example:

Given:

Two points = (2, 4) and (5, 6)

Required: m

Solution:

First let what to be the first and second point. Whether we let (2, 4) or (5, 6) as our first point, it doesn't matter because it will not affect the answer.

Let:

P1(2,4)
P2(5,6)

m = (y_{2 -} y1) / (x_{2 -} x1)

m = (6 - 4) / (5 - 2)

m = 2/3

(Solution 2)

Let:

P1(5,6)
P2(2,4)

m = m = (y_{2 -} y1) / (x_{2 -} x1)

m = (4 - 6) / (2 - 5)

m = -2 / -3

m = 2/3

- We just got the same answer!

* Angle of Inclination

- It is the angle formed by a line with respect to the positive x-axis.

Formula:

tanϴ = m

Where:

ϴ = angle of inclination
m = slope of the line

Examples:

* Determine the angle of inclination of a line passing through (-3, -2) and (5, 1).

Given: 

P1(-3, -2)
P2(5, 1)

Required: ϴ

Solution:

tanϴ = m

ϴ = tan^-1 [(y_{2 -} y1) / (x_{2 -} x1)]

ϴ = tan^-1 [(1 - (-2)) / (5 - (-3))] 

ϴ = tan^-1 [(1 + 2) / (5 + 3)] 

ϴ = tan^-1(3/8)

ϴ = 20.36 degrees

* Find the slope of the line having an angle of inclination equal to 45 degrees.

Given:

ϴ = 45 

Required: m

Solution:

tanϴ = m

m = tan(45 degrees)

m = 1

* The equation of the line is 3x - y + 4 = 0. Find its angle of inclination.

Given:

3x - y + 4 = 0

Required: ϴ

Solution:

Make in the form:

y = mx + b

3x - y + 4 = 0

y = 3x + 4

tanϴ = m

ϴ = tan^-1(3)

ϴ = 71.57 degrees

* Angle between Two Lines

    - angle formed by two intersecting lines.

Formula:

ø = ϴ₂ - ϴ₁

tanø = (tanϴ2 - tanϴ₁) / (1 + tanϴ₁tanϴ2)

tanø = (m₂ - m1) / (1 + m1m₂)


Where:

ϴ₁ = angle of the first line
ϴ ₂ = angle of the second line

m1 = slope of the first line
m2 = slope of the second line
ø = angle between the two lines

Note:

    - If you get a negative angle just get its absolute. Perhaps, m₂ and m₁ have just been interchanged.

Example:

* Two intersecting lines have slopes of m = 2 and m = 3. Find the angle between them.

Given:

Let:

m₁ = 2
m₂ = 3

Required: ø

Solution:

tanø = (m₂ - m1) / (1 + m1m₂)

tanø = (3 - 2) / (1 + (2)(3)) 

ø = tan^-1(1/7)

ø = 8.13 degrees

Distance Between Two Points, Division and Midpoint of Line Segment - Analytic Geometry

* Distance between Two Points

- if line 1 and line 2 are parallel with x and y-axis respectively, x₂ minus x₁ will be the distance parallel to the x-axis while y₂ minus y₁ will be the distance parallel to the y-axis. We will then form a right triangle whose hypotenuse is the distance between the two points :( x₁, y₁) and (x₂, y₂).

Formula:  
(Derived from Pythagorean Theorem) 

Where:

 
d = distance between the two points.
x₁ = x-coordinate of point 1
x₂ = x-coordinate of point 2
y₁ = y-coordinate of point 1
y₂ = y-coordinate of point 2

Examples:

* Find the distance between the points (1,3) and (5,3).

Given:

Let:
P1(1,3)
P2(5,2)

Required: d

Solution:

d = √[(x₂ - x1)2 - (y₂ - y1)2]

d = √[(5 - 1)2 - (2 - 3)2]

d = 4.12 units

* Division of a Line Segment

            - Simply finding the coordinates of the points that divides a line segment into parts (ratio).

Formula:

x = x₁ + r(x₂ - x₁)
y = y₁ + r(y₂ - y₁)

Where:

x and y = coordinate of the point that divides the line segment.
x₁ and x₂ = x-coordinates of the line segment's end points
y₁ and y₂ = y-coordinates of the line segment's end points
r = ratio

Example:

* Find the coordinates of the points that divide the line into 3 parts. The end points of the line are (1, 3) and (5, 8)

Given:
Let:
P1(1,3)
P2(5,8)

Required: points that divides the line (A and B)

Solution:

(solving for 1st point: A)

* ratio: 1/3 

x_A = x₁ + r(x₂ - x₁)

= 1 + (1/3)(5 - 1)

= 1 + 1.33

x_A = 1.33

y_A = y₁ + r(y₂ - y₁)

= 3 + (1/3)(8 - 3)

= 3 + 1.67

y_A= 4.67


A(2.33,4.67)

* ratio: 2/3

x_B = x₁ + r(x₂ - x₁)

= 1 + (2/3)(5 - 1)

= 1 + 2.67

x_B= 3.67

y_B = y₁ + r(y₂ - y₁)

= 3 +(2/3)(8 - 3)

= 3 + 3.33

y_B = 6.33

B(3.67,6.33)

Note: 
- If you are dividing a line. The number of points is always less than the number of division.
 

    no. points = no. division - 1

    e.g.
    no. divisions = 5
    no. points = 4

- to check if your answer is correct, solve the distance of each part of the line segment by using distance formula. The sum of the distances of the parts must be equal with the distance between the endpoints of the line segment itself.

* Midpoint of a Line Segment
            - a point that divides a line segment into two equal parts.

Formula:

x_M = (x_{2 +} x1)/2

y_M = (y_{2 +} y1)/2

Where:

x_M and y_M = coordinates of midpoint
x₁ and x₂ = x-coordinates of the endpoints of the line segment.
y₁ and y₂ = y-coordinates of the endpoints of the line segment.

Example:

* Find the midpoint of the line. Endpoints: (3,4) and (4,6)

Given:
Let:
P1(3,4)
P2(4,6)

Required: M

Solution:

x_M = (x_{2 +} x1)/2

= (3 + 4)/2 

= 7/2

x_M = 3.5



y_M = (x_{2 +} x1)/2

= (4 + 6)/2

= 10/2

y_M = 5

M(3.5,5)