Have you ever wondered how we measure the steepness of a hill / the inclination of a ramp? How about calculating velocities in physics / determining the rate of change in a graph? The answer lies in a fundamental concept of geometry called slope.
It is essential to understand slope to unlock the secrets hidden within mathematical equations and the world around you. Understanding slope enables us to solve problems requiring motion, growth, / change, design and build structures with precise inclinations, comprehend graphs and data representations, and analyze the gradients of hills and slopes in real-world scenarios.
In this article, we will explore the definition, types, methods, applications, and examples of slope in geometry.
What is Slope in Geometry?
In geometry, a slope is a fundamental concept that is used to find the inclination of the line as well as the steepness of the line and different objects like hills. The term slope is a numerical value that finds the relation between two points on the line.
The two points on the line would be the x-coordinates and y-coordinates on the line. It is denoted by “m”. The slope of the line can also be defined as the ratio of vertical change (change in y coordinates) to the horizontal change (change in x coordinates) between two points on the line.
The above definition of the slope can be written mathematically:
Slope = change in y coordinates : change in x coordinates
Where
- change in y coordinates = y2 – y1
- change in x coordinates = x2 – x1
The slope of the line equation will be:
Slope = (y2 – y1) / (x2 – x1)
The slope of the line is the best way to analyze whether the line is steep or gentle. The steeper and gentle line depends on the values of the slope such as the lower slope value denotes a gentle slope while the larger value of the slope denotes a steeper slope.
The slope of the line also lets us know the rate of change of the line. A smaller value of slope indicates a slower rate of change while a larger slope value indicates a faster rate of change. Hence, the rate of change of the line could be slower and faster depending on the values of the slope. You can find the slope of the line with the help of a slope finder to get results in few seconds with the graph.
Types of Slope
Four basic types of slope are essential in measuring the rate of change of the line and the steepness of the line. These are:
- Positive Slope
- Negative Slope
- Zero Slope
- Undefined Slope
Here we’ll explore the above types of slopes with their significance.
Positive Slope
It is the type of slope that indicates the upward inclination of the line. The positive slope rises from left to right. The increase in the values of the x-axis causes an increase in the values of the y-axis. The rising of the line will be faster if the positive slope is steeper.
Such as a hill gradually increases from left to right in ascending order from the ground. The growth, progress, and upward trends will occur as the result of a positive relation between x-coordinate values and y-coordinate values.
Negative Slope
It is the type of slope that indicates the downward inclination of the line. The negative slope goes downhill from left to right. The increase in the values of the x-axis causes a decrease in the values of the y-axis. The down hilling of the line will be faster if the negative slope is steeper.
Such as a hill gradually decrease from left to right in descending order from the top. The descent, decrease, and downward trend will occur as the result of a negative relation between x-coordinate values and y-coordinate values.
Zero Slope
It is the type of slope that indicates the horizontal line that neither rises nor descends. In the zero slope, the value of the y coordinate remains the same even if we occur a change in the x coordinates.
There is no change occurring in this type of slope for a dependent variable with respect to the independent variable. The flat surface and moving a car with a constant speed are examples of this kind of slope.
Undefined Slope
It is the type of slope that indicates the vertical line that neither rises nor descends. In the undefined slope, the value of the x coordinate remains the same even if we occur a change in the y coordinates.
There is no change occurring in this type of slope for an independent variable with respect to the dependent variable. The tall building is an example of this kind of slope.
How to find the slope of a line?
Example 1: Positive slope
Find the steepness of the line with the help of the given points of x coordinates and y coordinates of the line.
(x1, y1) = (2, 4) and (x2, y2) = (30, 8).
Solution
Step 1: First of all, write the x coordinates and y coordinates points of the line.
x1 = 2, y1 = 4, x2 = 30, y2 = 8
Step 2: Now take the rise over the run formula of the slope of the line.
Slope of a line = m = 𝚫Y/𝚫X = y2 – y1 / x2 – x1
Step 3: Now calculate the change in the values of the x coordinate and the change in the values of the y coordinate.
For 𝚫X
𝚫X = X2 – X1
𝚫X = 30 – 2
𝚫X = 28
For 𝚫Y
𝚫Y = Y2 – Y1
𝚫Y = 8 – 4
𝚫Y = 4
Step 4: Substitute the values of change in the x coordinate and the change in the y coordinate in the formula.
Slope of a line = m = 𝚫Y/𝚫X
Slope of a line = m = 4/28
Slope of a line = m = 2/14
Slope of a line = m = 1/7
Example 2: Negative slope
Find the steepness of the line with the help of the given points of x coordinates and y coordinates of the line.
(x1, y1) = (6, 40) and (x2, y2) = (14, 8).
Solution
Step 1: First of all, write the x coordinates and y coordinates points of the line.
x1 = 6, y1 = 40, x2 = 14, y2 = 8
Step 2: Now take the rise over the run formula of the slope of the line.
Slope of a line = m = 𝚫Y/𝚫X = y2 – y1 / x2 – x1
Step 3: Now calculate the change in the values of the x coordinate and the change in the values of the y coordinate.
For 𝚫X
𝚫X = X2 – X1
𝚫X = 14 – 6
𝚫X = 8
For 𝚫Y
𝚫Y = Y2 – Y1
𝚫Y = 8 – 40
𝚫Y = -32
Step 4: Substitute the values of change in the x coordinate and the change in the y coordinate in the formula.
Slope of a line = m = 𝚫Y/𝚫X
Slope of a line = m = -32/8
Slope of a line = m = -16/4
Slope of a line = m = -4
Example 3: zero slope
Find the steepness of the line with the help of the given points of x coordinates and y coordinates of the line.
(x1, y1) = (-12, 6) and (x2, y2) = 18, 6).
Solution
Step 1: First of all, write the x coordinates and y coordinates points of the line.
x1 = -12, y1 = 6, x2 = 18, y2 = 6
Step 2: Now take the rise over the run formula of the slope of the line.
Slope of a line = m = 𝚫Y/𝚫X = y2 – y1 / x2 – x1
Step 3: Now calculate the change in the values of the x coordinate and the change in the values of the y coordinate.
For 𝚫X
𝚫X = X2 – X1
𝚫X = 18 – (-12)
𝚫X = 18 + 12
𝚫X = 30
For 𝚫Y
𝚫Y = Y2 – Y1
𝚫Y = 6 – 6
𝚫Y = 0
Step 4: Substitute the values of change in the x coordinate and the change in the y coordinate in the formula.
Slope of a line = m = 𝚫Y/𝚫X
Slope of a line = m = 0/30
Slope of a line = m = 0
A slope calculator can be used to find the positive, negative, or zero slope with steps to avoid above manual calculations.
Final Words
The slope of the line is a fundamental concept in geometry that is essential for various fields of study and daily life. We’ve discussed the basics of the slope in geometry with its solved examples.