Verified Answer :

  • Define the slope as the ratio of the vertical change to the horizontal change, \(m = \frac{\Delta y}{\Delta x}\).
  • Identify the given changes: \(\Delta y = 2\) and \(\Delta x = 4\).
  • Substitute the values into the formula to get \(m = \frac{2}{4}\).
  • Simplify the fraction to \(\frac{1}{2}\), which is option B.
  • Final Answer: \(\boxed{B}\)

Explanation

  1. Understanding Slope

The slope of a line is a measure of its steepness and direction. It is defined as the ratio of the vertical change, often called the "rise," to the horizontal change, often called the "run," between any two points on the line. Mathematically, for a change in coordinates from \((x_1, y_1)\) to \((x_2, y_2)\), the slope \(m\) is calculated as:
\[m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}.\]
In this problem, we are given the displacement of the line: a movement of 4 units to the right (a positive horizontal change) and 2 units upward (a positive vertical change).

  1. Calculating and Simplifying the Slope

From the problem description, we have:

  • The horizontal change, or \(\Delta x\), is \(4\).
  • The vertical change, or \(\Delta y\), is \(2\).

Applying these values to the slope formula, we get:
\[m = \frac{\Delta y}{\Delta x} = \frac{2}{4}.\]
By dividing both the numerator and the denominator by their greatest common divisor, which is 2, we simplify the fraction:
\[m = \frac{2 \div 2}{4 \div 2} = \frac{1}{2}.\]
Therefore, the slope of the line is \(\frac{1}{2}\). Looking at the options provided, this corresponds to option B.

Examples

In real-life engineering and construction, the slope represents the grade or pitch of a ramp, road, or roof. For example, if a wheelchair ramp rises 2 feet over a horizontal distance of 4 feet, its slope is \(\frac{2}{4} = 0.5\), or a 50% grade. Understanding how to calculate this ratio helps ensure safety and compliance with accessibility standards in architecture.

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