Analytical Geometry aims its studies through the conciliation between Algebra and Geometry. In this way, some situations can be methodically analyzed, through geometric interpretation and algebraic relations.
One of these important relationships in Analytical Geometry is the distance between a point and a straight line in the Cartesian plane.
The distance between a point and a line is calculated by joining the point to the line through a segment, which must form a right angle with the line (90º). To establish the distance between the two we need the general equation of the line and the coordinate of the point. The following figure establishes the graphic condition of the distance between point P and the line r, with the segment PQ being the distance between them.
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Establishing the general equation of the line s: ax + by + c = 0 and the coordinate of the point P(x0yy0), we were able to arrive at the expression capable of calculating the distance between the point P and the line s:
d = |ax0 + by0 + c|
√(the2 + b2)
This expression arises from a generalization made, and can be used in situations that involve the calculation of the distance between any point and a straight line.
Example
given the point A(3, -6) and r: 4x + 6y + 2 = 0. Establish the distance between A and r using the expression given above.
We have to:
x: 3
y: -6
to: 4
b: 6
c: 2 
by Mark Noah
Graduated in Mathematics
Brazil School Team
Analytical Geometry - Math - Brazil School
Would you like to reference this text in a school or academic work? Look:
SILVA, Marcos Noé Pedro da. "Distance between point and line"; Brazil School. Available in: https://brasilescola.uol.com.br/matematica/distancia-entre-ponto-reta.htm. Accessed on June 28, 2021.
