Perpendicular lines
Perpendicular Lines covers the relationship between the slopes of two perpendicular lines: their slopes are negative reciprocals, meaning m₁ × m₂ = -1. This concept is taught in Yoshiwara Elementary Algebra Chapter 3: Graphs of Linear Equations and helps Grade 6 students identify, verify, and construct perpendicular lines algebraically. An important special case is when one line is horizontal (slope 0) and the other is vertical (undefined slope).
Key Concepts
Property Two lines are perpendicular if the product of their slopes is $ 1$, that is, if $$m 1 m 2 = 1$$ or if one of the lines is horizontal and one is vertical. The slope of one perpendicular line is the negative reciprocal of the other: $$m 2 = \frac{ 1}{m 1}$$.
Examples A line with slope $m 1 = 5$ is perpendicular to a line with slope $m 2 = \frac{1}{5}$ because their product is $5 \times ( \frac{1}{5}) = 1$. The lines $y = \frac{3}{4}x + 2$ and $y = \frac{4}{3}x 1$ are perpendicular. Their slopes are $m 1 = \frac{3}{4}$ and $m 2 = \frac{4}{3}$, and their product is $(\frac{3}{4})( \frac{4}{3}) = 1$. The line $y = 5$ is horizontal (slope is 0), and the line $x = 1$ is vertical (slope is undefined). These lines are perpendicular.
Explanation Perpendicular lines intersect at a perfect right angle, like the corner of a book. Their slopes are opposites in two ways: one is positive while the other is negative, and their fractions are flipped (reciprocals).
Common Questions
What is the relationship between slopes of perpendicular lines?
The slopes of perpendicular lines are negative reciprocals of each other: m₁ × m₂ = -1. For example, slopes of 2 and -1/2 are perpendicular.
How do you find the slope of a line perpendicular to a given line?
Take the negative reciprocal of the given slope. Flip the fraction and change the sign.
Are horizontal and vertical lines perpendicular?
Yes. A horizontal line has slope 0 and a vertical line has an undefined slope, and they meet at right angles.
Where are perpendicular lines covered in Yoshiwara Elementary Algebra?
Perpendicular lines are in Chapter 3: Graphs of Linear Equations of Yoshiwara Elementary Algebra.
How do you verify if two lines are perpendicular?
Multiply their slopes. If the product is -1, the lines are perpendicular. If the slopes cannot be multiplied (one is undefined), check if one is horizontal and one is vertical.