## Tuesday, December 16, 2014

### Circle and Secant Segment Problems, 2

Category: Plane Geometry

"Published in Newark, California, USA"

From the given figure, find the ratio x : y.

 Photo by Math Principles in Everyday Life

Solution:

Consider the given figure above

 Photo by Math Principles in Everyday Life

The given figure consists of two secant segments that are drawn in a circle from an external point. In the same figure, there are two chords that intersect inside a circle. In this case, we can solve for the values of x and y by using the two theorems.

If the first theorem says "The measure of an angle formed by two chord that intersect inside a circle is equal to the half the sum of the measures of the intercepted arcs.", then the working equation is

If the second theorem says "The measure of an angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside a circle is equal to half the difference of the measures of the intercepted arcs.", then the working equation is

From the two working equations, if we add the two equations, y will be eliminated. The value of x is

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Substitute the value of x to either one of the working equations, we have

Therefore, the ratio of x and y is