Rise Over Run

Rise over run is the inclination of a line in the coordinate axis. The rise is computed along the vertical y-axis and the run is computed along the horizontal x-axis and the rise over run is expressed as a ratio, where the rise along the y-axis is divided by the run along the x-axis. The rise over run is also referred to as the slope or the gradient of the line.

Rise over run for a line is calculated using the formulas m = (y₂ - y₁)/(x₂ - x₁) = Tanθ = f'(x) = dy/dx. Let us learn more in detail about each of the formulas of rise over run, and its applications, with the help of Examples, FAQs.

Rise Over Run is the inclination of the line with respect to the vertical axis and the horizontal axis. This measure helps in knowing the proportional relationship of the line along the vertical y-axis and horizontal x-axis of the coordinate axis. The rise over run is also commonly referred to as the slope of the line and is represented with the alphabet m.

For a line passing through the points \((x_1, y_1)\), and \((x_2, y_2)\), the rise of \(y_2 - y_1\) is measured vertically along the y-axis, and the run of \(x_2 - x_1\) is measured horizontally along the x-axis. And the rise over run for the line connecting these points is \(\dfrac{(y_2 - y_1)}{(x_2 - x_1)}\).

Rise over run is referred to as the ratio, as it divides the rise with the run. Thus the rise over run ratio is an important metric to define the line in the coordinate system. Sometimes rise over run ratio is also termed as the gradient of the line, or slope of the line.

The rise over run can be calculated through different methods. The rise over run can be calculated using the points on the line, the angle made by the line, or the differentiation of the curve or line. Let us check in detail each of the methods of computing the rise over run.

From Two Points: The points on the line are used to find rise over run of the given line. For any two given points \((x_1, y_1)\), and \((x_2, y_2)\), the difference of the y - coordinates \(y_2 - y_1\) represent the rise, and the difference of the x-coordinate \(x_2 - x_1\) represent the run. The rise over run can be calculated using the following formula.

Rise Over Run = m = \(\dfrac{(y_2 - y_1) }{(x_2 - x_1)}\)

Angle Made By The Line: The angle made by a line with the positive x-axis in the anticlockwise direction, θ is useful to find the rise over run of the given line. The tan of the angle θ is used to compute the rise over run of the line.

Rise Over Run = m = Tanθ

Differentiation: The differentiation of the function f(x), which represents the equation of a line can be used to compute the rise over run ratio. The function y = f(x) is a function with an equation of the independent variable x, and the dependent variable y. Here we find the differentiation of y with respect to x, which is represented as follows.

Rise Over Run (m) = f'(x) = dy/dx

One of the above three methods can be used based on the available inputs, to find the rise over run ratio. Further, the rise over run is always computed with reference to the x-axis.

Rise over run is often referred to as the slope of the line, and it has numerous applications in coordinate geometry. The following are some of the important applications of rise over run.

Relation Between Two Lines: The rise over run of two-line helps us to know if the two lines are parallel, perpendicular or inclined to each other. The product of the slope or the rise over run ratio of two perpendicular lines is equal to -1, and the rise over run ratio of two parallel lines is equal. Also knowing the slopes of two lines helps in knowing the angle between the two lines. The angle θ between two lines having slopes of m₁ and m₂ is as follows.

Tanθ = \(\dfrac{n_1 - m_2}{1 + m_1.m_2}\)

Relation with the Coordinate Axes: The rise over run ratio helps in knowing the placement of the line with respect to the coordinate axes. The rise over run value of 1 indicates that the line is equally inclined to both the axes. Further, the value lesser than 1 indicates that it is move inclined to x-axis and the run part of the ratio is large. And if the value is more than 1, then it indicates that the line is more inclined to the y-axis, and the rise is more than the run.

Equation of a Line: The equation of a line can be formed only after knowing its slope or the rise over run ratio. The equation of a line using the point-slope form, slope-intercept form, required the use of slope or rise over run ratio. For the given slope m, the possible formulas to find the equation of a line are:

y - y₁ = m(x - x₁)

y = mx + c

With this, we can understand the importance of the rise over run for a line, which served numerous applications in coordinate geometry.

☛Related Topics

• Slope
• Point Slope Form
• Slope Intercept Form
• x and y-intercept
• X and Y Coordinates
• Coordinate Plane

What Is Rise Over Run?

Rise over run is the inclination of the line with respect to the coordinate axis. The rise is the measure along the vertical y-axis, and the run is the measure along the horizontal x-axis. The rise over run is the ratio of the y component and the x-component of the line. The rise over run of a line passing through the points. \((x_1, y_1)\), and \((x_2, y_2)\), is referred as m = \(\dfrac{(y_2 - y_1) }{(x_2 - x_1)}\)

What Is The Other Name For Rise Over Run?

Rise over run is often referred to as the slope of the line or the gradient of the line. It is the measure obtained by dividing the y component of the line with the x component of the line. The rise over run ratio is an important characteristic to define the line, and is required to find the equation of the line.

What Is Rise Over Run Ratio?

The rise over run is expressed as a ratio. The rise over run ratio for a line is computed using any points \((x_1, y_1)\), and \((x_2, y_2)\), on the line. Here the difference of the y - coordinates \(y_2 - y_1\) represent the rise, and the difference of the x-coordinate \(x_2 - x_1\) represent the run. The rise over run ratio can be calculated using the following formula.

Rise Over Run = m = \(\dfrac{(y_2 - y_1) }{(x_2 - x_1)}\)

How Do You Calculate Rise Over Run Ratio?

The rise over run ratio is obtained by dividing the y component of the line with the x component of the line. The rise over run is also called the slope or the gradient of the line and can be computed using one of the following three methods.

• The rise over run for a line passing through the points \((x_1, y_1)\), and \((x_2, y_2)\), is m = \(\dfrac{(y_2 - y_1) }{(x_2 - x_1)}\)
• The angle made by a line with the positive x-axis in the anticlockwise direction, θ is useful to find the rise over run for the given line. Rise Over Run = m = Tanθ
• The differentiation of the function f(x), which represents the equation of a line can be used to compute the rise over run ratio. The function y = f(x) is differentiated to get rise over run value of (m) = f'(x) = dy/dx

What Are The Uses Of Rise Over Run Ratio?

The rise over run is used to find the equation of the line: the equations of point-slope form - y - y₁ = m(x - x₁), and the slope intercept form - y = mx + c requires m, the slope or the rise over run ratio. The rise over run helps to easily understand the inclination of the line with respect to each of the coordinate axes.

Can The Rise Over Run Be Negative Value?

The rise over run value can also be a negative value. If the change in the y component and the x component are in inverse proportion, then we have a negative value for the rise over run ratio.