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

[SB257426]

Im talking about this formula:

View attachment 40165


I was doing a hunters hill high extension 2 trial and they asked to find the min value of |z|. Would I need to derive this formula or can i just use it in an exam?

 

[SB257426]

here is the formula...

 

[SadCeliac]

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

 

[SB257426]

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

yeah it is ....


i thought as well but they used that formula to find the min value of z

 

[SadCeliac]

yeah it is ....


i thought as well but they used that formula to find the min value of z

what is the context? send the question???

 

[SB257426]

yeah it is ....


i thought as well but they used that formula to find the min value of z

if u look at hunters hill ext 2 trial if its on here then have a look at 11 D)

 

[synthesisFR]

View attachment 40166

here is the formula...

iirc thats the formula for perpendicular distance. i forgot how to derive it or idk if we even did derive it, last time i did it was year ten.
usually for short dist i just do projections or dot product

 

[Luukas.2]

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

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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]

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.