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 

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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,

Circle and Secant Segment Problems, 4

Category: Plane Geometry

"Published in Newark, California, USA"

A circle can be drawn through points X, Y, and Z. 
a. What is the radius of the circle?
b. How far is the center of the circle from point W?

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Solution:

Consider the given figure above

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Expand the line segment YW from point W and label the other point as point V.

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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 is

Hence, the length of XZ is 20 and YV is 22. The midpoint of XZ is 10 units from X and the midpoint of YV is 11 units from Y. From their midpoints, draw a vertical and a horizontal line to locate their intersection which is point C as the center of a circle as follows

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From the figure, by using Pythagorean Theorem, the distance of C from W is

and the radius of a circle which is the distance of C from Z is

 

 

 

 

If you will get the distance of C from X, Y, or V, the value of length must be the same otherwise point C is not the center of a circle.

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Circle and Secant Segment Problems, 3

Category: Plane Geometry

"Published in Newark, California, USA"

PT is tangent to circle O. Secant BA is perpendicular to PT at P. If TA = 6 and PA = 3, find (a) AB, (b) the distance from O to AB, and (c) the radius of circle O.

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Solution:

Consider the given figure above

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Since PT is perpendicular to AP, then we can use Pythagorean Theorem in order to solve for the length of PT as follows

Next, let's analyze further the given figure as follows

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If you draw a line segment from point O which is perpendicular to AB, then AB will be bisected. Also, the perpendicular line is the distance of a chord or AB to the center of a circle.

If a theorem says "When two secant segments are drawn to a circle from an external point, the product of one secant segment and its external segment equals the product of the other secant segment and its external segment.", then the working equation is
 

Substitute the values of the line segments in order to solve for the value of x as follows
 

 

 

 

 

 
Therefore, the length of a chord is
 

 

 

 
If OT is perpendicular to PT and and PT is perpendicular to PB, then OT is parallel to PB. In this case, the distance of AB to the center of a circle is
 

 

 
and the radius of a circle is