Does anyone have a derivation of the formula for the shortest distance to a line? (1 Viewer)

SadCeliac

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is this from point P to a line AB??? I thought we do that with projections?
 

Luukas.2

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The perpendicular distance formula gives the shortest distance from a point to the line , which is


It was a part of the 2 unit syllabus until the change in 2020. Now the only way is to derive the formula before using it or use vectors / projections for MX1 and MX2 students.

It can be derived as follows: Let X and Y be the points on the line such that PX and PY are parallel to the x- and y-axes, respectively. Let Q be the point on the line that is closest to P. The result follows by setting PQ = d (the perpendicular distance sought) and finding the area of triangle XPY in two different ways.
 

Luukas.2

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Another derivation: Solving the equations of the circle and the line simultaneously should yield a quadratic equation, which has only one solution if the line is a tangent to the circle. By setting the discriminant to zero, the result should follow by making the radius (which is also the perpendicular distance sought) the subject.
 

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