Problem 54 - College Algebra with Modeling and Visualization

Question

Find an equation of the line satisfying the conditions. Perpendicular to \(x=15\), passing through \((1.6,-9.5)\)

Step-by-Step Solution

Answer

The equation is \(y = -9.5\).

Step 1: Understand Perpendicularity to Vertical Lines

The line given as \(x = 15\) is a vertical line. A line that is perpendicular to a vertical line must be a horizontal line.

Step 2: Horizontal Line Equation

A horizontal line has an equation of the form \(y = c\), where \(c\) is the \(y\)-coordinate of any point the line passes through.

Step 3: Use the Given Point

Since the horizontal line must pass through the point \((1.6, -9.5)\), substitute \(-9.5\) for \(c\) in the equation \(y = c\).

Step 4: Write the Equation

The equation of the line that is perpendicular to \(x = 15\) and passes through \(1.6, -9.5\) is \(y = -9.5\).

Key Concepts

Perpendicular Lines

Perpendicular lines are lines that intersect at a right angle, which is equal to 90 degrees. When discussing perpendicular lines in coordinate geometry, you need to identify the types of lines involved. One important case is when you have vertical and horizontal lines.

• Vertical lines have equations of the form \(x = a\), where \(a\) is the constant \(x\)-value for the line.
• Horizontal lines have equations of the form \(y = c\), with \(c\) being the constant \(y\)-value for the line.

When a line is perpendicular to a vertical line, the resulting perpendicular line must be horizontal, as only a horizontal line can intersect a vertical line at a 90-degree angle.
This understanding helps solve problems like finding the line that is perpendicular to a given vertical line. Knowing the geometry of perpendicular lines simplifies the process as you can directly write the equation of the line using the point through which it passes.

Horizontal Lines

Horizontal lines are special lines in geometry characterized by having a constant \(y\)-value everywhere along the line. Their equation is often written as \(y = c\), where \(c\) is a constant.

• A horizontal line does not slope upwards or downwards, it just extends from left to right, parallel to the \(x\)-axis.
• In terms of slope, a horizontal line has a slope of zero.

If you want to find the equation of a horizontal line and you know a point the line passes through, simply take the \(y\)-coordinate of the point and that becomes \(c\) in the equation \(y = c\).
For instance, if a line passes through the point \((1.6, -9.5)\), the equation of this horizontal line is \(y = -9.5\). This shows it runs straight through \(-9.5\) on the \(y\)-axis.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, combines algebraic principles with geometric concepts to describe shapes and their properties in a coordinate plane.

• Each point in a coordinate plane is represented by a pair \((x, y)\), where \(x\) and \(y\) are the coordinates.
• Equations of lines are expressed in terms of these coordinates, allowing us to find intersections, distances, and angles.

When dealing with problems involving lines, you often use coordinate geometry to translate geometric descriptions into algebraic equations. For vertical and horizontal lines, this usually involves using simple forms like \(x = a\) or \(y = c\).
Coordinate geometry is crucial for understanding how different types of lines interact on the plane, such as identifying when they are parallel, intersecting, or perpendicular. It allows you to solve problems by visualizing points and lines, instead of relying solely on abstract reasoning.