Circle and Secant Segment Problems, 7

Category: Plane Geometry

"Published in Newark, California, USA"

In the given diagram, CD is a tangent, arc AC ≅ arc BC, AB = 3, AF = 6, and FE = 10. Find ED and CD. (Hint: Let ED = x and CD = y. Then write two equations in x and y.)

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Solution:

Consider the given figure above
 

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Draw a line segment from C to E to form ΔFCE and ΔCED and draw a line segment A to C to form ΔABC and ΔACE and label the further figure as follows
 

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Since ∠AFB ≅ ∠CFE because they are vertical angles and the sides of two triangles are proportional to each other, then ΔFCE and ΔAFB are similar triangles. In short, the two triangles formed from the intersection of two line segments in a circle are always similar. If they are similar triangles, then we can solve for the measurement of CE as follows
 

 

 

 

Same thing with the line segments BF and FC as follows

  

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, line segment BF is
 

 

 

If arc AC ≅ arc BC, then line segments AC and BC are also congruent which is equal to 10 + 6 = 16.

Consider ΔAFB,

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By Cosine Law, the value of ∠AFB which is also equal to
∠CFD is
 

 

 

 

 

 

 

Hence, the value of ∠AFC which is also equal to ∠BFE is

Consider ΔACE,

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By Cosine Law, the value of ∠CAE is

If a theorem says "The measure of an inscribed angle is equal to half the measure of its intercepted arc.", then the working equation is

 

 

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

 

 

 

 

 

 

 

If a theorem says "The measure of an angle formed by a chord and a tangent is equal to half the measure of the intercepted arc.", then the working equation is
 

 

 

By Sine Law at ΔFCD, we can have the first working equation as follows
 

 

 

 

 

Since there are two variables in a working equation above, then we need another working equation in order to solve for the values of x and y. 

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 other working equation is
 

 

 

but


Hence, the above equation becomes

By using Quadratic Formula, the value of x is

 

Choose the positive sign because all sides of a triangle must be positive, we have
 

 

 

 

 

Substitute the value of x to either of the two equations in order to solve for the value of y, we have

Therefore, ED = 1.863672 and CD = 5.769925.  

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

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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

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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 

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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,