Problem 53 - A Graphical Approach to Precalculus with Limits

Question

Find the equation of the line described, giving it in slope-intercept form if possible. Perpendicular to \(x=3,\) passing through \((1,2)\)

Step-by-Step Solution

Answer

The equation is \(y = 2\).

Step 1: Understand the Given Line

The line given is described by the equation \(x = 3\), which is a vertical line and does not have a defined slope. Any line perpendicular to a vertical line must be horizontal.

Step 2: Identify the Slope of the Perpendicular Line

If a line is perpendicular to a vertical line, it must be horizontal. A horizontal line has a slope of 0.

Step 3: Use the Slope-Intercept Form

The slope-intercept form of a line is given by \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. For a horizontal line with slope 0, the equation simplifies to \(y = b\).

Step 4: Substitute the Given Point into the Equation

The line passes through the point \((1,2)\). Since the line is horizontal, the y-coordinate of every point on the line is the same. Therefore, the equation of our line is \(y=2\).

Key Concepts

Perpendicular Lines

When we refer to lines being perpendicular to one another, they intersect at a right angle or 90 degrees. A perpendicular relationship between lines has specific characteristics:

• If one line has a slope (m) of a particular value, the perpendicular line will have a slope that is the negative reciprocal of this value. This means if the original slope is \(m \), then the perpendicular line will have a slope of \(-\frac{1}{m}\).
• For vertical lines, which have an undefined slope, the concept of a negative reciprocal doesn't apply. Instead, vertical lines are always perpendicular to horizontal lines since they meet at 90-degree angles.
• In our problem, the given line is vertical with an equation \(x = 3\). Therefore, the perpendicular line must be horizontal.

Understanding these relationships allows us to find and write equations for lines based on their geometric relationship to one another. In this manner, recognizing perpendicularity simplifies finding equation forms, like slope-intercept form, which is dependent on correctly recognizing line slopes.

Horizontal Line

A horizontal line is characterized by having a constant y-value across all its points. This results in a slope of 0, because the rise over run ratio (change in y over change in x) is 0 for any two points on the line:

• The mathematical representation of a horizontal line in slope-intercept form is \(y = b\), where \(b\) is the y-value shared by all points along the line.
• Horizontal lines run parallel to the x-axis and never intersect it. Therefore, they do not change as the x-value changes, hence no vertical change.
• In the original exercise, because the line is perpendicular to \(x = 3\) (a vertical line), it must be horizontal. Given the point \((1,2)\), the line will simply be \(y = 2\).

Understanding horizontal lines facilitates solving problems related to parallel and perpendicular slopes, as they are often involved in creating perpendicular pairs with vertical lines.

Vertical Line

Vertical lines provide an interesting scenario in geometry as they take the form of \(x = a\), where \(a\) is a specific x-value shared by all points on this line. Here are some key points:

• Vertical lines have an undefined slope since their rise can be any value yet the run is always 0, making the slope calculation impossible (division by zero).
• These lines run parallel to the y-axis, indicating that the x-value remains constant across all points, thus signifying no horizontal change.
• In our original problem, the vertical line is \(x = 3\). Any line perpendicular to this, to maintain the right angle, must be a horizontal line (as explained in the Perpendicular Lines section).

Recognizing and understanding vertical lines are crucial for various tasks in geometry, especially when coupled with finding slopes and constructing perpendicular line equations in slope-intercept form.