Concentric Circles Diameter

Question

Let $C_1$ and $C_2$ be concentric circles such that the diameter of $C_1$ is $2	ext{ cm}$ longer than that of $C_2$. If a chord of $C_1$ has length $6	ext{ cm}$ and is a tangent to $C_2$, then the diameter, in cm of $C_1$ is

Accepted Answer

10. Let radius of $C_1$ be $R$ and radius of $C_2$ be $r$. Diameter of $C_1$ is $2	ext{ cm}$ longer than $C_2 \implies 2R - 2r = 2 \implies R - r = 1$. Length of chord of $C_1$ tangent to $C_2$ is $6	ext{ cm}$. By Pythagoras theorem: $$R^2 - r^2 = 3^2 = 9$$ $$(R - r)(R + r) = 9$$ Since $R - r = 1$, we get $R + r = 9$. Adding $R - r = 1$ and $R + r = 9$ gives $2R = 10$. Diameter of $C_1 = 2R = 10	ext{ cm}$.

Answer

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Concentric Circles Diameter

Explanation

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