Definition: Two lines are said to be perpendicular if they intersect or meet at a right angle, which is 90 degrees. If you think about the letter “T”, where the top bar meets the vertical bar is a perfect example of perpendicular lines.
Examples:
- The edges of a book on a table. The cover of the book is horizontal, while the pages stand vertically. The angle between them? You guessed it! 90 degrees.
- Street intersections. Think of a crosswalk where one road meets another directly in the middle. Those roads are usually perpendicular to each other.
Visual Representation: Imagine drawing a horizontal line on a piece of paper. Now, draw a vertical line that crosses the horizontal line. The point where they cross is called the “point of intersection”. If the angle at that point is 90 degrees, then the two lines are perpendicular.
Symbol: When we want to say that line AB is perpendicular to line CD, we write it as AB ⊥ CD.
Questions learners might have:
- Can curved lines be perpendicular to straight lines?
- Generally, when we speak of perpendicular lines, we’re referring to straight lines. However, a tangent to a curve at a point can be perpendicular to a radius drawn to the same point on the curve (as seen with circles).
- Are all vertical lines perpendicular to all horizontal lines?
- Yes! Anytime a vertical line meets a horizontal line, they form a right angle of 90 degrees, making them perpendicular.
- Can two lines be perpendicular in a 3D space?
- Absolutely! The concept extends to three-dimensional space as well. Think of the corner of a room where the two walls meet the floor.
Problems to solve:
- Problem 1: Given the equations of two lines: y = 3x + 2 and y = -1/3x – 1. Determine if they are perpendicular.
Solution: To determine if the lines are perpendicular, check the slopes. Lines are perpendicular if their slopes are negative reciprocals of each other. Slope of the first line: 3 Slope of the second line: -1/3 Since 3 and -1/3 are negative reciprocals, the lines are perpendicular.
- Problem 2: Are the lines represented by 2x – 5y = 10 and 5x + 2y = 15 perpendicular?
Solution: To determine if the lines are perpendicular, find the slopes.
- For the first equation, rearrange to get y = mx + b. The slope, m, for the first line is 2/5.
- For the second equation, rearrange to get y = mx + b. The slope, m, for the second line is -5/2. Since 2/5 and -5/2 are negative reciprocals, the lines are indeed perpendicular.
- Problem 3: Draw a right triangle on the board. Ask the students: Are the two shorter sides of the triangle perpendicular to each other?
Solution: Yes! In a right triangle, the two shorter sides meet at a right angle, making them perpendicular.
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