Circle Inscribed in a Trapezoid Problems

Category: Plane Geometry

"Published in Newark, California, USA"

Find the exact area of the given trapezoid in the figure:

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Solution:

Consider the given figure above

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Since the given figure is an isosceles trapezoid, then it follows that ∠A ≅ ∠B, ∠C ≅ ∠D, and AD ≅ BC. If a circle is inscribed in an isosceles trapezoid, then its radius is tangent to the sides of an isosceles trapezoid. Let's analyze and label further the given figure as follows
 

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Line segment EG that passes the center of a circle bisects the two bases of an isosceles trapezoid. Line segment OB bisects ∠B and line segment OC bisects ∠C. 

Consider rt. ∆OGB and rt. ∆OFG. If OG ≅ OF and OB ≅ OB, then it follows that BG ≅ BF. In this case, BF = 9.

Consider rt. ∆OEC and rt. ∆OFC. If OE ≅ OF and OC ≅ OC, then it follows that EC ≅ CF. In this case, CF = 4.

Hence, the length of CB = CF + FB = 9 + 4 = 13.

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The value of x is 

By Pythagorean Theorem, the altitude of an isosceles trapezoid is

 

 

 

 

Therefore, the area of an isosceles trapezoid is
 

 

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.