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Chapter 8: Problem 77

Answer true or false. A vertical line is always perpendicular to a horizontal line.

Short Answer

Expert verified
True.

Step by step solution

01

Understand the Definitions

A vertical line is a line that goes straight up and down. A horizontal line is a line that goes straight across from left to right. Perpendicular lines are two lines that intersect at a right angle (90 degrees).
02

Analyze the Geometric Setup

Visualize a vertical line, which would be like the line connecting the floor to the ceiling. A horizontal line would be like the line connecting opposite walls. Imagine they intersect at one point. Because one is up-and-down and the other is side-to-side, the angle they form is exactly 90 degrees.
03

Apply the Perpendicular Criteria

Since a vertical line and a horizontal line meet to form four 90-degree angles (also known as right angles), they meet the condition for being perpendicular lines.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertical Line
A vertical line is incredibly important in geometry as it aligns perfectly with the direction of gravity. Imagine standing still and looking up towards the sky or down towards your feet; the imaginary line you are following is vertical. In terms of coordinates in a graph, a vertical line is represented as all the points where the x-coordinate is the same. For instance, the line represented by the equation \( x = 3 \) is vertical. These lines are parallel to the y-axis, which extends vertically on a graph.
  • Direction: Up and down
  • Graph Representation: \( x = \text{constant} \)
  • Aim: Parallel to the Y-axis
Horizontal Line
A horizontal line runs parallel to the horizon. Whenever you look straight ahead from one side to the other, your view traces a horizontal path. Horizontally oriented structures can help provide stability and balance, much like the horizon line in drawings. On a graph, a horizontal line is shown by all the points where the y-coordinate remains constant. For example, the line \( y = 4 \) is a horizontal line. It runs parallel to the x-axis, extending side-to-side across a graph.
  • Direction: Side-to-side
  • Graph Representation: \( y = \text{constant} \)
  • Aim: Parallel to the X-axis
Perpendicular Lines
Perpendicular lines are a fundamental concept in geometry that adds structure and rigidity to two-dimensional space. Two lines are considered perpendicular if they intersect at a right angle, precisely 90 degrees. This concept can often be visualized as the intersection between a vertical and a horizontal line.

Imagine the corner of a piece of paper; the edges meet at a right angle, demonstrating perpendicularity. When looking at graphs, two lines are perpendicular if the product of their slopes is \(-1\). However, vertical and horizontal lines are naturally perpendicular because vertical lines have undefined slopes, while horizontal lines have a slope of zero.
  • Angle of Intersection: 90 degrees
  • Visual Example: Corner of a square
  • Slope Relationship: Product of slopes is \(-1\), or vertical intersects horizontal

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Most popular questions from this chapter

Answer true or false. A vertical line is always parallel to another vertical line.

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