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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
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
