Problem 30 - Algebra and Trigonometry

Question

Find an equation of the line that satisfies the given conditions. Through \((2,6) ;\) perpendicular to the line \(y=1\)

Step-by-Step Solution

Answer

The equation of the line is \(x = 2\).

Step 1: Identify the characteristics of the given line

The given line is \(y = 1\). This is a horizontal line, which means it is parallel to the x-axis.

Step 2: Determine the slope of the line perpendicular to the given line

Since the line \(y = 1\) is horizontal, its slope is 0. A line that is perpendicular to a horizontal line has an undefined slope, which is a vertical line. Vertical lines are parallel to the y-axis.

Step 3: Write the equation of the vertical line through the given point

A vertical line passing through a specific point \((x_0, y_0)\) has an equation of the form \( x = x_0 \). Given the point \((2, 6)\), the equation of the line is \( x = 2 \).

Key Concepts

Perpendicular Lines

When you have two lines that meet at a right angle, they are known as perpendicular lines. These lines cross each other at an angle of 90 degrees. Understanding perpendicularity can help in solving various geometry and algebra problems.
A unique feature of perpendicular lines is their slopes. If you have a non-vertical line, the slope of a line that is perpendicular to it is the negative reciprocal of the original line's slope.

• For example, if the slope of one line is \( m \), the slope of the line perpendicular to it will be \( -\frac{1}{m} \).

However, it's interesting to note that when dealing with horizontal and vertical lines, things change. Horizontal lines have a slope of zero, and vertical lines, which are their perpendicular counterparts, have an undefined slope. This makes them unique among perpendicular lines as described further below.

Horizontal and Vertical Lines

The concepts of horizontal and vertical lines are foundational in understanding geometry and coordinate systems.

• Horizontal lines run from left to right across the plane, staying parallel to the x-axis. Their equation typically looks like \( y = c \) where \( c \) is the constant y-value across the line.
• Vertical lines, on the other hand, run up and down the plane, always parallel to the y-axis, and have equations of the form \( x = k \), where \( k \) is the constant x-value.

One special property of these lines is how they relate to each other in terms of perpendicularity. A horizontal line and a vertical line are always perpendicular because they effectively form right angles where they intersect. This is particularly useful when finding lines perpendicular to given horizontal or vertical lines, just as in our exercise.

Slope

The slope of a line is a measure of its steepness and direction. It's usually denoted by \( m \) in algebraic equations and can be calculated as the "rise" over the "run" between two points on the line. In mathematical terms, if you have two points \((x_1, y_1)\) and \((x_2, y_2)\) on the line, the slope \( m \) is given by:\[ m = \frac{y_2 - y_1}{x_2 - x_1}\] This formula shows how much the line rises or falls as you move from one point to another horizontally.

• Horizontal lines have a slope of zero, as there's no vertical change.
• Vertical lines, however, have an undefined slope because they do not run horizontally at all.

Understanding the concept of slope is crucial, especially in determining the relationships between different lines, such as parallel and perpendicular lines. In particular, knowing when a slope is undefined helps identify vertical lines, which was key in solving the original exercise.