Problem 46 - Contemporary Precalculus

Question

Determine whether the line through \(P\) and \(Q\) is parallel or perpendicular to the line through \(P=(3,3), Q=(-3,-1)\) and \(R=(2,-2), S=(4,-5)\)

Step-by-Step Solution

Answer

Question: Determine if the line through points P(3,3) and Q(-3,-1) is parallel or perpendicular to the line through points R(2,-2) and S(4,-5). Answer: The line through P and Q is neither parallel nor perpendicular to the line through R and S.

Step 1: Calculate the slope of the line through P and Q

To calculate the slope, use the formula \((y2 - y1) / (x2 - x1)\). For points P\((3,3)\) and Q\((-3,-1)\), the slope can be determined by \(((-1) - (3)) / ((-3) - (3))\).

Step 2: Simplify the slope of the line through P and Q

Simplify the previous expression: \((-1 - 3) / (-3 - 3) = (-4) / (-6)\). The slope is simplified as \(2/3\).

Step 3: Calculate the slope of the line through R and S

To calculate the slope, use the same formula for points R\((2, -2)\) and S\((4, -5)\). We get the slope: \(((-5) - (-2)) / ((4) - (2))\).

Step 4: Simplify the slope of the line through R and S

Simplify the expression: \((-5+2) / (4 - 2) = (-3) / (2)\). The slope is simplified as \(-3/2\).

Step 5: Determine if the lines are parallel or perpendicular

Compare the slopes of the lines. They are neither equal (not parallel) nor negative reciprocals of each other (not perpendicular). Therefore, the line through P and Q is neither parallel nor perpendicular to the line through R and S.

Key Concepts

Slope

When understanding the concept of slope in linear equations, think of it as a measure of how steep a line is. The slope is a number that describes the direction and steepness of a line. It's calculated as the "rise over run," or more specifically, the change in vertical distance (rise) divided by the change in horizontal distance (run) between two points on the line.

The formula for slope, often denoted as "m," is \[m = \frac{y_2 - y_1}{x_2 - x_1}\]where

• \(x_1, y_1\):
  The coordinates of the first point
• \(x_2, y_2\):
  The coordinates of the second point

The slope can be positive, negative, zero, or undefined:

• **Positive slope**: Line rises as it moves from left to right.
• **Negative slope**: Line falls as it moves from left to right.
• **Zero slope**: Horizontal, flat lines.
• **Undefined slope**: Vertical lines, as the run is zero.
  

Understanding slope is essential as it forms the foundation for more complex geometrical concepts like parallel and perpendicular lines.

Parallel Lines

Parallel lines are lines that never intersect, no matter how far they're extended on a plane. When we talk about the slopes of parallel lines, the key property is that parallel lines have the same slope. This means that in the equation for slope, the coefficient of the x-term (which is the slope, \(m\)) will be identical for both lines.

For example, if two lines have equations given as \[y = m_1x + c_1\] and \[y = m_2x + c_2\], then \(m_1\) should equal \(m_2\) for these lines to be parallel.

A quick way to verify that two lines are parallel is simply to compare their slopes. If they are equal, the lines are parallel. It's important to note that parallel lines are equidistant from one another at all points, and thus they never meet or cross each other. This property plays a crucial role in many geometric proofs and applications.

Perpendicular Lines

Perpendicular lines are lines that intersect at a right angle (90 degrees). The slopes of perpendicular lines have a special relationship: they are negative reciprocals of each other. This means that if one line has a slope \(m\), then the line perpendicular to it will have a slope of \(-1/m\).

For example, if a line has a slope of \(2/3\), the line perpendicular to it will have a slope of \(-3/2\). This relationship between the slopes ensures that the angle formed by their intersection is exactly 90 degrees.

To determine if two lines are perpendicular, multiply their slopes. If the product is \(-1\), the lines are perpendicular. This rule is incredibly useful when analyzing geometric figures or solving problems involving right angles. The concept of perpendicularity is central to numerous branches of mathematics, including Euclidean geometry and trigonometry.