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

 

Proving of Two Intersecting Circles

Category: Plane Geometry

"Published in Newark, California, USA"

A line is tangent to two intersecting circles at P and Q. The common chord is extended to meet PQ at T. Prove that T is the midpoint of PQ.

Solution:

To illustrate the problem, it is better to draw the figure as follows

Photo by Math Principles in Everyday Life

Let A and B are the centers of two circles and CD is the common chord of two circles. If CD is extended to meet PQ at T, then CT is the common external segment of the two circles. 

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 A is
 

 

and for circle B is
 

 

Since CT is the common external segment of two circles, then we can equate the two working equations as follows

Take the square root on both sides of the equation, we have

Since PT ≅ TQ, then T is the midpoint of PQ.
 

Circle and Secant Segment Problems, 5

Category: Plane Geometry

"Published in Newark, California, USA"

Draw two intersecting circles with common chord PQ and let X be any point on PQ. Through X draw any chord AB of one circle. Also draw through X any chord CD of the other circle. Prove that AX • XB = CX • XD.

Solution:

In the given word problem, you can draw two intersecting circles of any size you wish and then label further the figure while analyzing the word problem as follows 

Photo by Math Principles in Everyday Life

If a theorem says "When two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other chord.", then the working equation for the small circle is

and for the big circle is

Since PQ is the common chord of two circles, then we can equate the two working equations as follows

Therefore,