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2009 AMC 12B Problem 9

Problem 9 of 25EasierGeometry

Triangle ABCABC has vertices A=(3,0),A = (3, 0), B=(0,3),B = (0, 3), and C,C, where CC is on the line x+y=7.x + y = 7. What is the area of △ABC?\triangle ABC?

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Solution

Line ABAB has equation x+y=3,x + y = 3, which is parallel to x+y=7,x + y = 7, so the area is independent of where CC lies on that line. Take C=(7,0).C = (7, 0). Then the base AC=4AC = 4 lies on the xx-axis with height 3,3, giving area 12⋅4⋅3=6.\dfrac{1}{2} \cdot 4 \cdot 3 = 6. Thus, the correct answer is A.

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Concepts: coordinate geometry · triangle area · parallel lines

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