Perpendicularity Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Perpendicularity.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Lines, segments, or planes that intersect at exactly a right angle of 90° to each other.

The corner of a book or a room—the two edges meet at precisely 90°.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Perpendicular slopes are negative reciprocals: m_1 \times m_2 = -1.

Common stuck point: Perpendicular slopes are negative reciprocals: if one slope is m, the other is -1/m. Product = -1.

Sense of Study hint: Try multiplying the two slopes together. If the product is exactly -1, the lines are perpendicular.

Worked Examples

Example 1

easy
Line \ell_1: y = 2x + 1. Write the equation of a line \ell_2 perpendicular to \ell_1 that passes through (4, 3).

Solution

  1. 1
    Step 1: Slope of \ell_1: m_1 = 2.
  2. 2
    Step 2: Perpendicular slope: m_2 = -\dfrac{1}{m_1} = -\dfrac{1}{2} (since m_1 \times m_2 = -1).
  3. 3
    Step 3: Point-slope form: y - 3 = -\dfrac{1}{2}(x - 4) \Rightarrow y = -\dfrac{1}{2}x + 5.

Answer

y = -\dfrac{1}{2}x + 5
Perpendicular lines meet at 90°. Their slopes are negative reciprocals: if one slope is m, the other is -1/m. This ensures m_1 \times m_2 = -1.

Example 2

medium
Determine whether triangle A(0,0), B(4,0), C(4,3) is a right triangle, and if so identify the right angle vertex.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Line m passes through (0, -1) with slope \dfrac{3}{5}. Write the equation of the line perpendicular to m through the origin.

Example 2

hard
Find the foot of the perpendicular from point P(3, 7) to the line \ell: y = 2x - 1.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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