How Do You Calculate Perpendicular?

Calculating perpendicularity primarily involves understanding the relationship between the slopes of lines or the properties of angles within geometric figures. Here's a breakdown:

Perpendicular Lines and Slopes

Two lines are perpendicular if they intersect at a right angle (90 degrees). The most common way to determine if lines are perpendicular is to examine their slopes.

• The Rule: Two lines are perpendicular if and only if the product of their slopes is -1. This can also be expressed as the slopes being "negative reciprocals" of each other.

• Finding the Perpendicular Slope:

  1. Identify the slope of the first line. Let's call it m₁.
  2. Calculate the negative reciprocal: The slope of the line perpendicular to the first line, m₂, is m₂ = -1/m₁. This means you flip the fraction and change the sign.
  • Example 1: If m₁ = 2 (or 2/1), then m₂ = -1/2.
  • Example 2: If m₁ = -3/4, then m₂ = 4/3.
  • Example 3: If m₁ = 5/2, then m₂ = -2/5.

• Horizontal and Vertical Lines:

  • A horizontal line has a slope of 0. A line perpendicular to a horizontal line is a vertical line, which has an undefined slope.
  • A vertical line has an undefined slope. A line perpendicular to a vertical line is a horizontal line, which has a slope of 0.

Perpendicular in Geometric Shapes

Perpendicularity is also important in geometry:

• Right Angles: Perpendicular lines, segments, or rays form right angles (90° angles).
• Altitude of a Triangle: The altitude of a triangle is a line segment from a vertex perpendicular to the opposite side (or the extension of the opposite side).
• Distance from a Point to a Line: The shortest distance from a point to a line is along the perpendicular segment from the point to the line.
• Squares and Rectangles: Sides that meet at a corner in squares and rectangles are perpendicular.

Example: Finding the Equation of a Perpendicular Line

Suppose you have a line with the equation y = -4/5 x + 2, and you want to find the equation of a line perpendicular to it that passes through the point (4, -3).

1. Find the perpendicular slope: The original slope is -4/5, so the perpendicular slope is 5/4.

2. Use point-slope form (or slope-intercept): We know the slope (m = 5/4) and a point (x = 4, y = -3). We can use slope intercept form: y = mx + b. Plug in m, x, and y.

3. Solve for the y-intercept (b):
   -3 = (5/4) * 4 + b
   -3 = 5 + b
   b = -8

4. Write the equation: The equation of the perpendicular line is y = 5/4 x - 8.