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Three tangent circles of radius 10 cm are drawn. All centers lie on the line AB. The tangent AC to the right-hand circle is drawn, intersecting the middle circle at D and E. Find the length of the segment DE.
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| Photo by Math Principles in Everyday Life |
Solution:
Consider the given figure above
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| Photo by Math Principles in Everyday Life |
Since AC is tangent to the third circle at C, then the radius of third circle is perpendicular to AC. Hence, the three lines becomes a right triangle as follows
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| Photo by Math Principles in Everyday Life |
By Pythagorean Theorem, the length of AC is
At the center of second circle, draw a line which is perpendicular to DE. Extend that line at the opposite ends of second circle as follows
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| Photo by Math Principles in Everyday Life |
If SQ is perpendicular to DE and passes thru the center of the circle P, then SQ bisects DE. Since RP is perpendicular to AC and CO is perpendicular to AC, then RP is parallel to CO. Because of this ∆ARP and ∆ACO are similar.
If ∆ARP and ∆ACO are similar, then we can solve for RP by ratio and proportion 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 for the second circle is
Therefore, the length of DE is
