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2007 AMC 12B Problem 6

Problem 6 of 25EasierAlgebraGeometry

Triangle ABCABC has side lengths AB=5,AB=5, BC=6,BC=6, and AC=7.AC=7. Two bugs start simultaneously from AA and crawl along the sides of the triangle in opposite directions at the same speed. They meet at point D.D. What is BD?BD?

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Solution

The perimeter is 5+6+7=18,5+6+7=18, so each bug crawls 99 before they meet. The bug going A→B→CA\to B\to C reaches DD on side BC,BC, having traveled AB+BD=9.AB+BD=9. Since AB=5,AB=5, we get BD=4.BD=4. Thus, the correct answer is D.

More practice

Concepts: relative speed · perimeter

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