Problem 48 - College Algebra with Corequisite Support

Question

For the following exercises, use the descriptions of each pair of lines given below to find the slopes of Line 1 and Line 2 . Is each pair of lines parallel, perpendicular, or neither? Line 1: Passes through (-8,-55) and (10,89) Line 2: Passes through (9,-44) and (4,-14)

Step-by-Step Solution

Answer

The lines are neither parallel nor perpendicular.

Step 1: Identify Points for Line 1

We know that Line 1 passes through the points (-8, -55) and (10, 89). We will label these points as \((x_1, y_1) = (-8, -55)\) and \((x_2, y_2) = (10, 89)\).

Step 2: Calculate the Slope of Line 1

The slope of Line 1, \(m_1\), can be calculated using the formula for the slope \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substitute the coordinates from Step 1 into the formula: \[ m_1 = \frac{89 - (-55)}{10 - (-8)} = \frac{89 + 55}{10 + 8} = \frac{144}{18} = 8. \] The slope of Line 1 is \(8\).

Step 3: Identify Points for Line 2

We know that Line 2 passes through the points (9, -44) and (4, -14). We will label these points as \((x_1, y_1) = (9, -44)\) and \((x_2, y_2) = (4, -14)\).

Step 4: Calculate the Slope of Line 2

The slope of Line 2, \(m_2\), is calculated using the same slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substitute the coordinates from Step 3 into the formula: \[ m_2 = \frac{-14 - (-44)}{4 - 9} = \frac{-14 + 44}{4 - 9} = \frac{30}{-5} = -6. \] The slope of Line 2 is \(-6\).

Step 5: Determine if Lines are Parallel, Perpendicular, or Neither

Parallel lines have equal slopes, and perpendicular lines have slopes that are negative reciprocals of each other. Since \(m_1 = 8\) and \(m_2 = -6\), the lines are neither parallel nor perpendicular because the slopes are not equal and do not satisfy the conditions for perpendicularity.

Key Concepts

Slope of a Line

The slope of a line is a measure of its steepness and direction. It's an essential concept in algebra and coordinate geometry. The slope formula is expressed as \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where:

• \( (x_1, y_1) \) and \( (x_2, y_2) \) are two distinct points on the line.
• \( m \) represents the slope.

This formula calculates the change in \( y \) (vertical change) divided by the change in \( x \) (horizontal change) between two points. Let's explore why slope is essential:

• A positive slope means the line rises from left to right.
• A negative slope means the line falls from left to right.
• A slope of zero indicates a horizontal line, and an undefined slope indicates a vertical line.

In our exercise, for Line 1 with points (-8,-55) and (10,89), the slope \( m_1 \) is \( 8 \). For Line 2, with points (9,-44) and (4,-14), the slope \( m_2 \) is \(-6\). Understanding slopes helps in identifying the behavior and the direction of different lines on a graph.

Parallel and Perpendicular Lines

In geometry, understanding the relationship between lines is crucial, especially when determining if they are parallel or perpendicular. Parallel lines have the same slope, which means they never intersect.

• For two lines to be parallel, \( m_1 = m_2 \).

Perpendicular lines, on the other hand, intersect at a 90-degree angle, and their slopes have a special relationship.

• If two lines are perpendicular, their slopes are negative reciprocals. This means \( m_1 \times m_2 = -1 \).

Analyzing the slopes from our example:

• Line 1 has a slope \( m_1 = 8 \).
• Line 2 has a slope \( m_2 = -6 \).
• Since \( 8 eq -6 \) and \( 8 \times -6 eq -1 \), the lines are neither parallel nor perpendicular.

Understanding these concepts helps in designing and analyzing geometric and real-world scenarios, such as roads or construction angles.

Coordinate Geometry

Coordinate geometry, or analytic geometry, uses a coordinate plane to explore the relationships between points, lines, and shapes. This branch of geometry allows us to use algebra to study geometric properties, providing a bridge between these two mathematical areas.

• The coordinate plane consists of an x-axis (horizontal) and y-axis (vertical).
• Points are expressed as \((x, y)\) coordinates, where \(x\) is the position along the x-axis and \(y\) along the y-axis.
• Lines can be represented with equations like \( y = mx + b \), where \(m\) is the slope, and \(b\) is the y-intercept (the point where the line crosses the y-axis).

In the given exercise, understanding points and how they relate on a graph is essential. Using the slope formula, we find the steepness of the lines which runs through pairs of these points.The use of coordinate geometry simplifies the process of solving problems involving distance, midpoints, and shapes within the Cartesian plane.