Question

The lines \(\mathrm{x}=-1\) and \(\mathrm{y}=4\) are (1) perpendicular to each other (2) parallel to each other (3) neither parallel nor perpendicular to each other (4) None of these

Short answer

Answer: The lines are perpendicular to each other.

Step-by-step solution

Determine the slope of each line

To find the slope of each line, we need to write those lines in the slope-intercept form (y = mx + b), where m represents the slope. For the line x = -1: As there is no y term, this line is vertical; meaning, its slope is undefined. For the line y = 4: Since y = 4 is a horizontal line, its slope is 0. Having slopes for the two lines, we can now compare them and choose the appropriate answer.

Compare the slopes and find the relationship

We can determine the relationship between the two lines by comparing their slopes. Since the first line has an undefined slope, it's a vertical line. The second line has a slope of 0, which indicates that it's a horizontal line. Vertical and horizontal lines are perpendicular to each other. Therefore, the given lines are perpendicular.

Choose the correct option

Based on the analysis in step 2, we can see that the two given lines are perpendicular to each other. The answer is: (1) perpendicular to each other

Key concepts

Slope of Lines

In mathematics, the slope of a line is a measure of its steepness and direction. It is often denoted by the letter \(m\). The slope indicates how much the line rises or falls as it moves from left to right. To determine the slope, you can think about it as the "rise over run," which is the change in \(y\) divided by the change in \(x\): \[m = \frac{y_2 - y_1}{x_2 - x_1}\]Here, \((x_1, y_1)\) and \((x_2, y_2)\) are two distinct points on the line.
If the slope is positive, the line goes upwards; if it's negative, the line goes downwards.

• A line with a slope of \(0\) is perfectly horizontal, meaning it doesn’t rise or fall at all as it moves straight across.
• An undefined slope corresponds to a vertical line because the line doesn't move horizontally but only vertically.

Understanding the slope completely allows you to describe how lines relate to each other in terms of parallelism or perpendicularity.

Vertical and Horizontal Lines

Vertical and horizontal lines are fundamental elements in coordinate geometry.
A vertical line is defined by its position on the \(x\)-axis and does not vary in \(y\). For example, the line \(x = -1\) means every point on the line has an \(x\)-coordinate of \(-1\) but the \(y\)-coordinate can be any number, so it goes straight up and down. As such, the slope of this line is undefined because it involves dividing by zero in the slope formula.
In contrast, a horizontal line is parallel to the \(x\)-axis and has a constant \(y\)-value. For instance, the line \(y = 4\) means every point on this line has a \(y\)-coordinate of \(4\) but the \(x\)-coordinate can be any number, so it runs straight left to right. The slope of a horizontal line is \(0\) because the line doesn’t rise or fall as it moves.

• Vertical lines are perpendicular to horizontal lines.
• This geometric principle makes them important for defining right angles in coordinate geometry.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This subject connects algebra and geometry through graphs of lines and shapes. By using coordinate geometry, complex geometric problems can be solved using algebraic techniques.
One of the fundamental concepts in coordinate geometry is the idea of using points on a plane to define geometric objects like lines, curves, and polygons. Points are typically defined in a coordinate system using pairs \((x, y)\), where \(x\) is the horizontal position, and \(y\) is the vertical position.

• By plotting these points, you can graphically represent lines and explore their properties such as slope, orientation, and length.
• Lines can be classified based on their slopes — parallel lines have equal slopes, while perpendicular lines have slopes that are negative reciprocals of each other. Though notably, vertical and horizontal lines are exceptions to this rule.

Coordinate geometry provides a powerful visualization and computation framework that makes it easier to solve and visualize geometric concepts and problems.