Perpendicularity Math Example 4

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Example 4

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

Solution

  1. 1
    Step 1: Perpendicular slope: m⊥=−12m_\perp = -\dfrac{1}{2}. Perpendicular through PP: y−7=−12(x−3)⇒y=−x2+172y - 7 = -\dfrac{1}{2}(x-3) \Rightarrow y = -\dfrac{x}{2} + \dfrac{17}{2}.
  2. 2
    Step 2: Intersect with ℓ\ell: 2x−1=−x2+1722x - 1 = -\dfrac{x}{2} + \dfrac{17}{2}. Multiply by 22: 4x−2=−x+17⇒5x=19⇒x=1954x - 2 = -x + 17 \Rightarrow 5x = 19 \Rightarrow x = \tfrac{19}{5}.
  3. 3
    Step 3: y=2⋅195−1=38−55=335y = 2 \cdot \tfrac{19}{5} - 1 = \tfrac{38-5}{5} = \tfrac{33}{5}.

Answer

Foot of perpendicular: (195, 335)\left(\dfrac{19}{5},\, \dfrac{33}{5}\right).
The foot of the perpendicular is the closest point on ℓ\ell to PP. We write the perpendicular through PP using slope −1/2-1/2 and solve the resulting 2×22 \times 2 linear system.

About Perpendicularity

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

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