which of the following statements about perpendicular lines is true? a) perpendicular lines have a constant distance between them. b) perpendicular lines intercept at a 90 degree angle. c) there is no standard symbol for perpendicular lines. d) perpendicular lines do not exist in the same place.
The correct answer and explanation is:
Correct Answer: b) Perpendicular lines intersect at a 90 degree angle.
Perpendicular lines are defined as two lines that meet or cross each other to form a right angle, which measures exactly 90 degrees. This characteristic is the fundamental property of perpendicularity in geometry. It applies whether the lines are in two-dimensional or three-dimensional space. The intersection forms four right angles at the point of contact.
Option a is incorrect because it describes parallel lines. Parallel lines have a constant distance between them and never intersect. They remain equidistant from one another along their entire length, unlike perpendicular lines that must intersect at one specific point.
Option c is also incorrect. There is a standard symbol for perpendicular lines in geometry. The symbol ⊥ is used to represent perpendicularity. For example, if line AB is perpendicular to line CD, it is written as AB ⊥ CD.
Option d is incorrect because for two lines to be perpendicular, they must lie in the same plane and must intersect. If two lines are not in the same plane, they cannot be perpendicular; instead, they are referred to as skew lines in three-dimensional geometry.
Understanding perpendicular lines is essential in various fields such as architecture, engineering, and design, where creating strong and stable structures often depends on accurate right-angle intersections. In mathematics, the concept is also foundational in trigonometry and coordinate geometry, where slopes of perpendicular lines in a plane are negative reciprocals of each other. This means if one line has a slope of m, the other will have a slope of -1/m. Recognizing and applying the concept of perpendicularity is key to solving many geometric and algebraic problems.