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2009 AMC 10A Problem 12

Problem 12 of 25IntermediateAlgebraGeometry

In quadrilateral ABCD,ABCD, AB=5,AB = 5, BC=17,BC = 17, CD=5,CD = 5, DA=9,DA = 9, and BDBD is an integer. What is BD?BD?

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Solution

In △BCD,\triangle BCD, the triangle inequality gives 5+BD>17,5 + BD \gt 17, so BD>12.BD \gt 12. In △ABD,\triangle ABD, it gives 5+9>BD,5 + 9 \gt BD, so BD<14.BD \lt 14. The only integer with 12<BD<1412 \lt BD \lt 14 is BD=13.BD = 13. Thus, the correct answer is C.

More practice

Concepts: triangle inequality · bounding to limit cases

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