## Sunday, December 21, 2014

### Circle and Secant Segment Problems, 6

Category: Plane Geometry

"Published in Newark, California, USA"

In the given diagram, PT is tangent to circle O and PN intersects circle O at J. Find the radius of the circle.

 Photo by Math Principles in Everyday Life

Solution:

Consider the given figure above

 Photo by Math Principles in Everyday Life

As you can see from the figure, it is hard to solve for the radius of a circle. We have to do something in the given figure first. Let's extend the given line segment PN so that it will meet the other side of a circle at point R and then label further the given figure as follows

 Photo by Math Principles in Everyday Life

If a theorem says "When a secant segment and a tangent segment are drawn to a circle from an external point, the product of the secant segment and its external segment is equal to the square of the tangent segment.", then the working equation for circle O is

Hence, the value of y which is one-half of the chord or line segment JR is

and the value of x is

By Pythagorean Theorem, the value of d which is the perpendicular distance of a chord to the center of a circle is

Therefore, by Pythagorean Theorem also, the radius of a circle is