Problem 60 - Algebra 1

Question

Choose which lines are perpendicular. Line \(p\) passes through \((4,0)\) and \((6,4)\) Line \(q\) passes through \((0,4)\) and \((6,4)\) Line \(r\) passes through \((0,4)\) and \((0,0)\) A line \(p\) and line \(q\) (B) line \(p\) and line \(r\) (C) line \(q\) and line \(r\) (D) none of these

Step-by-Step Solution

Answer

(D) None of these

Step 1: Calculate the Slope of Line \(p\)

The slope of a line passing through points \((x_1, y_1)\) and \((x_2, y_2)\) can be found using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For line \(p\), plug in the values \((4,0)\) and \((6,4)\) into the formula to get: \(m_p = \frac{4-0}{6-4} = 2\)

Step 2: Calculate the Slope of Line \(q\)

Similarly, calculate the slope for line \(q\) which passes through \((0,4)\) and \((6,4)\). After substituting the values, we get: \(m_q = \frac{4-4}{6-0} = 0\)

Step 3: Calculate the Slope of Line \(r\)

Calculate the slope for line \(r\) which passes through \((0,4)\) and \((0,0)\). However, since the denominator in our slope calculation is zero (since \(x_2 - x_1 = 0-0 = 0\)), this means the line \(r\) is vertical and its slope is undefined.

Step 4: Identify Perpendicular Lines

The product of slopes of perpendicular lines is -1. By comparing the calculated slopes: \(m_p * m_q = 2 * 0 = 0\) and \(m_p * m_r = 2 * Undefined \) does not exists, also \(m_q * m_r = 0 * Undefined \) does not exist. Therefore, there are no pairs of lines that are perpendicular.

Key Concepts

Slope of a Line

Understanding the slope of a line is critical in analyzing the behavior of the line. The slope is a measure of how steep a line is. It's calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line.

In simpler terms, if you were climbing a hill represented by a line on a graph, the slope tells you how steep the climb would be. The value of the slope can be positive, negative, zero, or undefined. A positive slope means the line is ascending as it moves from left to right, while a negative slope indicates it's descending. A zero slope equals a flat, horizontal line and an undefined slope corresponds to a perfectly vertical line.

To visualize this, picture a ladder leaning against a wall. The angle of the ladder determines the slope. If the ladder is lying flat on the ground, it has a zero slope. If it stands upright, it has an undefined slope. In all other positions, the incline of the ladder represents different slopes, which can be computed using specific coordinates on the ladder and the ground.

Undefined Slope

A line with an undefined slope is one that goes straight up and down, or vertically. This characteristic occurs because the horizontal change between any two points on a vertical line is zero, and dividing by zero is a mathematical operation that's not defined.

In the context of our textbook exercise, line r is an example of a line with an undefined slope. Since it passes through the points

(0,4) and (0,0), when you apply the slope formula, you end up with a zero in the denominator, which causes the slope to be undefined. Conceptually, you can think of a vertical line as a wall; no matter how far you walk along the wall, you won't move forward or backward, hence the slope is undefined.

Slope Formula

The slope formula, a crucial tool in geometry and algebra, is used to find the steepness of a line. The typical expression of this formula is
\( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \( m \) represents the slope, and \((x_1, y_1)\) and \((x_2, y_2)\) are coordinates of two distinct points on the line.

Applying the slope formula helps detect relationships between lines, such as whether they are parallel, perpendicular, or neither. For instance, in the exercise provided, calculating the slopes of lines p, q, and r allowed us to compare them and determine that there were no perpendicular pairings among them. A common mistake when using the slope formula is not paying attention to the order of the points and the operations, which can lead to incorrect results.