Problem 47 - Algebra 1: Concepts and Skills

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 lines (p, q, and r) are perpendicular to each other.

Step 1: Finding the Slope of Line p

First, calculate the slope of line p. The formula for calculating the slope is \((y2-y1)/(x2-x1)\). Substituting the given points, the slope (\(m_p\)) of line p is \((4-0)/(6-4) = 2

Step 2: Finding the Slope of Line q

Next is to calculate the slope of line q, again by using the formula \((y2-y1)/(x2-x1)\). By substituting the coordinates given, the slope (\(m_q\)) of line \(q\) is \((4-4)/(6-0) = 0

Step 3: Finding the Slope of Line r

Let's find the slope of line \(r\) with the formula \((y2-y1)/(x2-x1)\). Putting in the given points, we find that the slope (\(m_r\)) of line \(r\) is \((4-0)/(0-0)\) which is undefined.

Step 4: Checking for Perpendicular Lines

Now, the criterion for perpendicularity is that the product of the slopes of the two lines should equal -1. This means we need to check if \(m_p * m_q = -1\), if \(m_p * m_r = -1\), or if \(m_q * m_r = -1\). In all cases, the product is not equal to -1. Thus, no pair of lines is perpendicular to each other.

Key Concepts

Slope of a Line

The slope of a line is a measure of its steepness. It is calculated using the formula \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]where

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

The slope tells us how much a line rises or falls as we move along it from left to right.
For example, if the slope is positive, the line rises as it moves right.
If negative, the line falls. A slope of zero means the line is horizontal, while an undefined slope corresponds to a vertical line.
In the given exercise, the slopes were calculated for lines \( p \), \( q \), and \( r \). Line \( p \) had a slope of 2, indicating an upward trend, line \( q \) was horizontal with a slope of 0, and line \( r \) was vertical with an undefined slope.

Criteria for Perpendicularity

Perpendicular lines intersect at a right angle (90 degrees).
One key property is that the product of their slopes is \[ m_1 \times m_2 = -1 \]This relationship is crucial for identifying perpendicular lines. For example:

• If one line has a slope \( m_1 \), a line perpendicular to it should have a slope \( m_2 \) such that their product is -1.
• If \( m_1 = 2 \), then \( m_2 \) must be \(-\frac{1}{2}\).

In our exercise, none of the line pairs had slopes that multiplied to -1. Thus, none were perpendicular.
This rule helps us quickly determine right-angle intersections without graphically measuring angles.

Coordinate Geometry

Coordinate geometry, or analytic geometry, is a branch of mathematics where geometric problems are solved using coordinates. It provides a way to describe geometric shapes in a numerical way, allowing us to calculate distances, midpoints, and slopes:

• Coordinates help us pinpoint the location of a point as \( (x, y) \) in a 2D plane.
• By using formulas, we can find various attributes of lines, such as slopes or lengths.

This field of study combines algebra and geometry to solve spatial problems efficiently.
In our exercise, using known points and coordinate calculations, we determined the qualities of lines \( p \), \( q \), and \( r \).
Such methods are invaluable for visually and mathematically analyzing geometric shapes and their relationships.