2012 AMC 10A Problem 11

Below is the professionally curated solution for Problem 11 of the 2012 AMC 10A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2012 AMC 10A solutions, or check the answer key.

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Concepts:tangent circles tangent line similarity

Difficulty rating: 1420

11.

Externally tangent circles with centers at points $A$ and $B$ have radii of lengths $5$ and $3,$ respectively. A line externally tangent to both circles intersects ray $AB$ at point $C.$ What is $BC?$

$4$

$4.8$

$10.2$

$12$

$14.4$

Solution:

Let $x$ be $BC.$ Note that $\triangle CEB$ and $\triangle CDA$ are similar due to angle-angle (tangent lines are perpendicular to radii).

Then $\dfrac{x}{3} = \dfrac{8 + x}{5}.$ Cross-multiplying gives us $5x = 24 + 3x$ $x = 12.$

Thus, D is the correct answer.

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