Problem 16 - Algebra 1: Concepts and Skills

Question

Determine whether the lines are perpendicular. $$ y=-5, x=5 $$

Step-by-Step Solution

Answer

Yes, the lines are perpendicular.

Step 1: Examination of the given lines

Here, two lines are provided: \(y=-5\) is a horizontal line and \(x=5\) is a vertical line.

Step 2: Checking for Perpendicularity

To check if the two lines are perpendicular, we need to verify if they intersect at a 90-degree angle. A vertical line (x=constant) is always perpendicular to a horizontal line (y=constant).

Key Concepts

Horizontal Line

A horizontal line in the coordinate plane is a straight line that stretches from left to right or right to left. It is characterized mainly by its equation of the form \( y = c \), where \( c \) is a constant. This equation indicates that no matter where you look along the line, the y-value remains the same.

• **Example**: The line \( y = -5 \) is a horizontal line. All points on this line have a y-coordinate of -5, such as (0, -5), (3, -5), and (-2, -5).
• **Properties**: A horizontal line never rises or falls; it stays consistent along the x-axis. It runs parallel to the x-axis.

Horizontal lines are very predictable and can aid in visualizing how they interact with other types of lines on a graph. Whether for coordinate graphs, geometry problems, or real-world scenarios like designing roads or layouts, understanding horizontal lines is crucial.

Vertical Line

A vertical line is a fundamental concept in geometry that runs up and down on a graph. It is represented by an equation of the form \( x = c \), where \( c \) stands for a constant value. This tells us that the x-value never changes, no matter the y-coordinate.

• **Example**: The line \( x = 5 \) is a vertical line. Each point on this line has an x-coordinate of 5, including (5, 0), (5, 3), and (5, -2).
• **Properties**: Vertical lines do not slant or curve; they extend indefinitely along the y-axis direction. They align parallel to the y-axis.

Vertical lines are essential when analyzing graphs in math and applied physics, as they often serve as boundary lines or axis limitations. Being comfortable with how vertical lines function helps when interpreting and solving geometric problems.

Intersection

An intersection refers to the point or set of points where two or more lines meet or cross each other. In graphing terms, it's where the lines have common coordinates and it's crucial in determining relationships such as perpendicularity or parallelism.

• **Perpendicular Lines**: When two lines intersect at a 90-degree angle, they are referred to as perpendicular. A horizontal line and a vertical line are often examples of perpendicular lines.
• **Finding Intersections**: To find where two lines intersect, you solve their equations simultaneously. For instance, the horizontal line \( y = -5 \) and vertical line \( x = 5 \) intersect at the point (5, -5).

Understanding intersections is vital in both pure and applied mathematics since it frequently solves optimization problems, determines areas of overlap, and explains how different geometric shapes relate to one another.