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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


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