Problem 29 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether the lines through each pair of points are perpendicular. $$(-1,-6)\( and \)(2,9) ;(-15,-1)\( and \)(5,3)$$

Step-by-Step Solution

Answer

The lines passing through the points (-1,-6) & (2,9) and (-15,-1) & (5,3) are not perpendicular.

Step 1: Compute the slope for each line

The slope \(m\) of a line passing through two points \( (x_1, y_1)\) and \( (x_2, y_2)\) can be found using the formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\) \n In this case, we have two pair of points. We first compute the slope \(m_1\) for the first pair of points \((-1,-6)\) and \((2,9)\) using the formula: \( m_1 = \frac{9 - (-6)}{2 - (-1)} = \frac{15}{3} = 5 \)\n Then we compute the slope \(m_2\) for the second pair of points \((-15,-1)\) and \((5,3)\) using the formula: \( m_2 = \frac{3 - (-1)}{5 - (-15)} = \frac{4}{20} = 0.2 \)

Step 2: Check if lines are perpendicular

Two lines are perpendicular if and only if the product of their slopes equals -1. This means we have to check if \( m_1*m_2 = -1\). In our case, \( m_1*m_2 = 5*0.2 =1 \). Therefore, the lines are not perpendicular.

Key Concepts

Slope of a Line

The slope of a line is a fundamental concept in coordinate geometry. It represents the rate at which a line rises or falls as you move along it. For any two points on a line, say \((x_1, y_1)\) and \((x_2, y_2)\), the slope \(m\) can be calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]This formula essentially measures the vertical change (rise) over the horizontal change (run) between the two points.

• Positive slope: Line rises from left to right.
• Negative slope: Line falls from left to right.
• Zero slope: Horizontal line.
• Undefined slope: Vertical line.

Determining the slope is crucial for understanding the direction and steepness of a line.

Point-Slope Formula

The point-slope formula is a powerful tool for writing the equation of a line when you know its slope and any point on the line. The formula is expressed as:\[ y - y_1 = m(x - x_1) \]Here, \((x_1, y_1)\) is a given point on the line, and \(m\) is the slope. This formula is particularly useful because:

• It allows you to quickly find the equation of a line.
• Helps in visualizing the line when given a point and a slope.
• Makes it easier to translate geometric problems into algebraic equations.

The point-slope formula is foundational for solving various problems in coordinate geometry and understanding the relationship between algebraic expressions and geometric lines.

Product of Slopes

The concept of the product of slopes is essential when determining if two lines are perpendicular. In coordinate geometry, two lines are perpendicular if the product of their slopes is equal to \(-1\). For example, if one line has a slope of \(m_1\) and another has a slope \(m_2\), then:\[ m_1 \times m_2 = -1 \]This principle stems from the fact that when two lines intersect at right angles, their slopes involve an opposite reciprocal relationship. Understanding the product of slopes is key:

• Helps identify perpendicular lines easily.
• Is a fundamental part of analyzing geometric figures in the coordinate plane.
• Applies to solve more complex problems involving right angles.

Remember, this check works only for non-vertical and non-horizontal lines (since their slopes can be undefined or zero).

Geometry in Coordinate Plane

Geometry in the coordinate plane is a method to study shapes, lines, and curves using algebraic equations and numerical calculations. This approach allows geometry to be analyzed using coordinates, providing a more mathematical perspective. Some key benefits and features include:

• Simple representation of geometric shapes such as lines and circles using equations.
• The ability to perform transformations (such as reflection and rotation) easily.
• Helps in solving geometric problems like finding intersection points or distances between shapes.

The coordinate plane combines both algebraic and geometric concepts, providing a tool to study various aspects like slope, distance, and angles in a systematic manner.