# Lesson Distance of a point from a given line in Cartesian Coordinate System

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 This Lesson (Distance of a point from a given line in Cartesian Coordinate System) was created by by hummingbird(0)  : View Source, ShowAbout hummingbird: In this lesson we will learn about the method of finding the perpendicular distance of a given point from a given line L1: Y = a.X + b There are several ways of finding the distance of a point from a line. I.e. using co-ordinate geometry, linear algebra and simple trigonometry. In this lesson, we will use trigonometric approach. In the above diagram, Given line is and the given point is . Now let us look at the construction of the triangle formation in order to obtain the perpendicular distance PC. Methodology We will first find the vertices of the triangle in order to get the side lengths and then by applying Sine Rule on triangle PAB and PBC we will calculate the desired distance PC. Step1 Draw a vertical line passing through the point P. This line L2: will cut the given line L1 at point A( ,). Similarly, draw a horizontal line passing through the point P. This line L3: will cut the given line L1 at point B(,). Now we need to calculate the vertices of the triangle PAB Step2 Now we have three lines with following equations: plugging in L1 will give us the point A and similarly plugging in L1 will give us the point B. Which can be calculated as: A( , ) B( , ) Step3 Calculate the length of the sides AP,PB and AB of a triangle by the simple distance formula in two-dimensional geometry. Distance formula : By this formula we can calculate the side lengths AP,PB and AB. Step4 Apply Sine rule on common angle B in triangle PAB and triangle PBC. Now, Lets plug the distance formula for AP,BP and AB in the above expression This is the final formula in terms of the given parameters xo, yo, a and b. Hence PC is the desired length of a point P from a line L1: Y = a.X + b Alternate Method This Method uses the concept of linear algebra and some advance results on co-ordinate geometry. The perpendicular length of a point from a line a*X + b*Y + c=0 is given by the formula Note: Here the line is of the form of a*X + b*Y + c=0. We can always change the given equation of line into this standard form and apply the above formula. Also look at the solver based on the above concept. Solver This lesson has been accessed 12149 times.