Skip to main content

2012 AMC 10A Problem 11

Problem 11 of 25IntermediateGeometry

Externally tangent circles with centers at points AA and BB have radii of lengths 55 and 3,3, respectively. A line externally tangent to both circles intersects ray ABAB at point C.C. What is BC?BC?

Answer choices

Show solution

Solution

Let xx be BC.BC. Note that △CEB\triangle CEB and △CDA\triangle CDA are similar due to angle-angle (tangent lines are perpendicular to radii). Then x3=8+x5. \dfrac{x}{3} = \dfrac{8 + x}{5}. Cross-multiplying gives us 5x=24+3x 5x = 24 + 3x x=12. x = 12. Thus, D is the correct answer.

More practice

Concepts: tangent circles · tangent line · similarity

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.