Inclined lines calculator

Line
(m_b)

Line input:

= 0

2 points

(x₁ , y₁):

(

, )

(x₂ , y₂)

(

, )

Point − Angle

(x_p , y_p)

(

, )

Angle =

Angle from line m_b

Point:

(

, )

Inclined

lines

Line 1

Line equation:

Slope (m₁)

Angle

Input line

Line equation:

Slope

Angle

Line 2

Line equation:

Slope (m₂)

Angle

Degree Radian

line geometry

Two lines intersection

Example1

Example2

Line rotated by an angle θ from a line

Figure 1

Figure 2

Angle α is called the inclination of a line L, and is defined in the range 0° ≤ α < 180°. A line parallel to the x-axis has an inclination of 0°, and any two parallel lines have the same inclination.

Because working directly with inclination in analytic geometry can be inconvenient, we instead use the slope m of the line, which is defined in terms of the inclination.

Notice that the tangent of the angle α is positive in the first quadrant (0° − 90°), and negative in the second quadrant (90° − 180°). It starts at 0 when α = 0 and increases to infinity as α approaches 90° (therefore, the slope m is not defined at this point). Beyond 90°, the tangent decreases from negative infinity and returns to 0 at 180°.

If two points along the line are known, then the slope can be calculated as:

m=tan⁡(α)=(y_2-y_1)/(x_2-x_(1 ) )=(y_1-y_2)/(x_1-x_(2 ) )

Note that subtracting the coordinates of point 1 from point 2 gives the same result as subtracting point 2 from point 1, since both the numerator and denominator change sign, leaving the ratio unchanged.

Line rotated by an angle ±θ from a given line whose slope is m_b

The slope of a given line is: m_b = tan θ_b or θ_b = tan^-1 m_b

The angles of the lines which are inclined by an angle θ from the original line are: θ₁ = θ_b − θ and θ₂ = θ_b + θ

Hence the slopes of the lines are:	m₁ = tan θ₁ = tan (θ_b − θ)
	m₂ = tan θ₂ = tan (θ_b + θ)

And the inclined lines equations which are passing through the point (x₀ , y₀) are:

y − y₀ = m₁(x − x₀) = tan(θ_b − θ)(x − x₀)

y − y₀ = m₂(x − x₀) = tan(θ_b + θ)(x − x₀)

Or expressed by m_b

y − y₀ = tan(tan^-1 m_b ± θ)(x − x₀) = tan(θ_b ± θ)(x − x₀)

the plus and minus signs stand for the two lines equations which lies either side of the original line.

Example 1 - Inclined lines.

find the equation of a line that is inclines by 25 degrees from the line given by the equation 5x −y + 3 = 0 and is passing through the point (-1 , -2)

Figure 1

The slope of the given line is:

m = 5

The angle of the given line is:

θ = tan^-1 5 = 78.7 degree

The angle of left incline line is:

θ₁ = 78.7 + 25 = 103.7 degree

The angle of right incline line is:

θ₂ = 78.7 − 25 = 53.7 degree

The slope of left incline line is:

m₁ = tan 103.7 = −4.1

The slope of right incline line is:

m₂ = tan 53.7 = 1.36

Hence the inclined lines equations at point (x₁ , y₁) are:

y = mx + (y₁ − mx₁)

y = −4.1x + [−2 + 4.1 (−1)] = −4.1x − 6.1

y = 1.36x + [−2 − 1.36 (−1)] = 1.36x − 0.64

Example 2 - Inclined lines.

find the equations of the lines that are inclined by 20 degrees to the line given by the equation 4y + 6 = 0 and is passing through the point where x = 5.