Question

A line passes through the points \(A(2,-3)\) and \(B(6,3)\) Find the slope of the line which is parallel to \(A B:\) (a) \(\frac{2}{3}\) (b) \(\frac{3}{2}\) (c) \(\frac{1}{2}\) (d) \(\frac{3}{4}\)

Short answer

Answer: (b) \(\frac{3}{2}\)

Step-by-step solution

Calculate the slope of line AB

To find the slope of the line AB, we will use the formula, slope = (y2 - y1)/(x2 - x1). In this case, point A has coordinates (x1 = 2, y1 = -3) and point B has coordinates (x2 = 6, y2 = 3). Now, we can plug these coordinates in the formula: Slope of AB = \(\frac{y2 - y1}{x2 - x1} = \frac{3 - (-3)}{6 - 2} = \frac{3 + 3}{6 - 2} = \frac{6}{4}\) Simplifying the fraction, we get: Slope of AB = \(\frac{3}{2}\)

Find the parallel line's slope

As parallel lines have the same slope, the slope of the line parallel to line AB will be the same as the slope of AB which we calculated in Step 1. Therefore, the slope of the line parallel to line AB is \(\frac{3}{2}\), which matches option (b). So, the correct answer is (b) \(\frac{3}{2}\).

Key concepts

Slope of a Line

The slope of a line is a measure of how steep the line is. It is calculated as the ratio of the change in the vertical direction (rise) to the change in the horizontal direction (run) between two points on the line. The formula for the slope is given by:\[\text{Slope} = \frac{y_2 - y_1}{x_2 - x_1}\]where \(x_1, y_1\) and \(x_2, y_2\) are the coordinates of any two distinct points on the line. In our exercise, points A and B are used to calculate the slope of line AB.

• The change in y (vertical change) is calculated by \(y_2 - y_1\).
• The change in x (horizontal change) is calculated by \(x_2 - x_1\).

For line AB, the slope is \(\frac{6}{4}\) which simplifies to \(\frac{3}{2}\). This value represents how much the line rises vertically for each unit of horizontal movement.
Understanding the slope is crucial for determining the direction and steepness of the line.

Parallel Lines

Parallel lines are lines in a plane that never meet. They are always the same distance apart and have the same slope. This means that regardless of where you measure the slope along the lines, they will have the same value.
For instance, if a line has a slope of \(\frac{3}{2}\), then any line parallel to it will also have a slope of \(\frac{3}{2}\).To find if two lines are parallel, simply compare their slopes:

• If the slopes are equal, the lines are parallel.
• If they differ, then the lines are not parallel.

Note: Parallel lines never intersect, no matter how far they are extended. This concept is important in geometry as well as various applications like engineering and architecture.

Equation of a Line

The equation of a line can be written in several forms, with the most common being the slope-intercept form, point-slope form, and standard form. For the purpose of our exercise, we'll focus on the slope-intercept form, which is generally more intuitive and straightforward. The slope-intercept form is:\[y = mx + c\]where:

• \(m\) represents the slope of the line.
• \(c\) is the y-intercept, which is the point where the line crosses the y-axis.

Knowing the slope from Line AB, the equation of any line parallel to it can be determined if a point on the new line is known.
This form is often used because it's easy to understand the line's behavior graphically. With both slope and intercept, you can quickly sketch the line and identify another point on it.