# Perpendicular from any Point of a Circle on a Diameter is a Mean Proportional

The perpendicular dropped from a point of a circle on a diameter is a mean proportional between the segments into which it divides the diameter.

Consider the equation of the circle with a center at the origin $O\left( {0,0} \right)$ is given by the equation

Let $P\left( {x,y} \right)$ be any point on the given circle. Let $AB$ be the diameter of the circle as shown in the given diagram. From $P$ draw a perpendicular on diameter $AB$ at $M$. The coordinates of $M$ are $M\left( {x,0} \right)$ because $M$ lies on the X-axis and $y = 0$ lies on the X-axis. The coordinates of $A$ and $B$ are $A\left( { - a,0} \right)$ and $B\left( {a,0} \right)$ respectively as shown in diagram.

Now we shall find the distance between $AM$ and $MB$ using the distance formula as follows:

Also

Now

From equation (i), we have ${y^2} = {r^2} - {x^2}$. Putting this value in equation (ii), we have

This shows that the perpendicular dropped from a point of a circle on a diameter is a mean proportional between the segments into which the diameter is divided.