Perpendicularity Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy
Line ℓ1:y=2x+1\ell_1: y = 2x + 1. Write the equation of a line ℓ2\ell_2 perpendicular to ℓ1\ell_1 that passes through (4,3)(4, 3).

Solution

  1. 1
    Step 1: Slope of ℓ1\ell_1: m1=2m_1 = 2.
  2. 2
    Step 2: Perpendicular slope: m2=−1m1=−12m_2 = -\dfrac{1}{m_1} = -\dfrac{1}{2} (since m1×m2=−1m_1 \times m_2 = -1).
  3. 3
    Step 3: Point-slope form: y−3=−12(x−4)⇒y=−12x+5y - 3 = -\dfrac{1}{2}(x - 4) \Rightarrow y = -\dfrac{1}{2}x + 5.

Answer

y=−12x+5y = -\dfrac{1}{2}x + 5
Perpendicular lines meet at 90°90°. Their slopes are negative reciprocals: if one slope is mm, the other is −1/m-1/m. This ensures m1×m2=−1m_1 \times m_2 = -1.

About Perpendicularity

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

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More Perpendicularity Examples

Example 2 medium

Determine whether triangle [formula], [formula], [formula] is a right triangle, and if so identify t

Example 3 easy

Line [formula] passes through [formula] with slope [formula]. Write the equation of the line perpend

Example 4 hard

Find the foot of the perpendicular from point [formula] to the line [formula].

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