- Identify the coordinates: \(P_1 = (92, -14)\) and \(P_2 = (8, -39)\).
- Apply the slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-39 - (-14)}{8 - 92}\).
- Simplify the fraction: \(\frac{-25}{-84} = \frac{25}{84}\).
- The final answer is \(\boxed{\frac{25}{84}}\).
Explanation
- Understanding the Problem and Data Analysis
The problem asks for the slope of a line passing through two distinct points in a 2D Cartesian plane. The coordinates of the points are given as \(P_1 = (92, -14)\) and \(P_2 = (8, -39)\). The slope \(m\) of a line that passes through two points \((x_1, y_1)\) and \((x_2, y_2)\) is defined as the change in the vertical coordinate (rise) divided by the change in the horizontal coordinate (run), which is expressed by the formula:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
Using the provided points, we identify \(x_1 = 92\), \(y_1 = -14\), \(x_2 = 8\), and \(y_2 = -39\). Our objective is to calculate this ratio and simplify it.
- Applying the Slope Formula and Calculation
We now substitute the values of the coordinates into the slope formula:
\[m = \frac{-39 - (-14)}{8 - 92}\]
First, we simplify the numerator, which represents the vertical change:
\[-39 - (-14) = -39 + 14 = -25\]
Next, we simplify the denominator, which represents the horizontal change:
\[8 - 92 = -84\]
Thus, the slope is given by the ratio of these values:
\[m = \frac{-25}{-84}\]
- Simplifying the Final Answer
Since both the numerator and the denominator are negative, the fraction simplifies to a positive value:
\[m = \frac{-25}{-84} = \frac{25}{84}\]
The fraction \(\frac{25}{84}\) cannot be simplified further because the greatest common divisor of 25 and 84 is 1. Therefore, the slope of the line passing through the points \((92, -14)\) and \((8, -39)\) is \(\frac{25}{84}\).
Examples
The concept of a slope is fundamental in many real-world applications, such as civil engineering or urban planning. For instance, if you are designing a ramp for a building to ensure it is accessible, the slope defines the steepness of the incline. If the ramp must rise 25 inches over a horizontal distance of 84 inches, the slope would be \(\frac{25}{84}\). This ratio helps professionals ensure that slopes are safe and meet building code requirements, providing a practical way to quantify steepness in everyday infrastructure.