In the figure, points $A$, $B$, $C$ and $D$ are on the same line and are the centers of four tangent circles at the same point. Segment $AB$ measures $8$ and segment $CD$ measures $4$. The circumferences woth centers $A$ and $C$ are of equal size. We know that the sum of the areas of the two medium circles is equivalent to the sum of the areas of the small and large circles. What is the length of segment $AD$?
Problem
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Tags: geometry
miyukina
26.03.2024 09:32
Difference of r between circle C and D is 4/2 = 2 Difference of r between circle A and B is 8/2 = 4 d^2 – c^2 = a^2 – b^2 = c^2 – b^2 2 × (c + d) = 4 × (c + b) c = d – 2b 2c^2 = d^2 + b^2 = 2 × (d – 2b)^2 = 2(d^2 – 4bd + 4b^2) 8bd = d^2 + 7b^2 0 = (d – b) × (d – 7b) Since d > b, we get d = 7b c = 5b so BC = 8 × (5 + 1) / (5 – 1) = 12 Answer = 8 + 12 + 4 = 24