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Square made by midpoints

Source: 2001 AMC-12 #20

December 4, 2005
analytic geometryrotationAMC

Problem Statement

Points A \equal{} (3,9), B \equal{} (1,1), C \equal{} (5,3), and D \equal{} (a,b) lie in the first quadrant and are the vertices of quadrilateral ABCD ABCD. The quadrilateral formed by joining the midpoints of AB‾,BC‾,CD‾, \overline{AB}, \overline{BC}, \overline{CD}, and DA‾ \overline{DA} is a square. What is the sum of the coordinates of point D D? <spanclass=′latex−bold′>(A)</span> 7<spanclass=′latex−bold′>(B)</span> 9<spanclass=′latex−bold′>(C)</span> 10<spanclass=′latex−bold′>(D)</span> 12<spanclass=′latex−bold′>(E)</span> 16 <span class='latex-bold'>(A)</span> \ 7 \qquad <span class='latex-bold'>(B)</span> \ 9 \qquad <span class='latex-bold'>(C)</span> \ 10 \qquad <span class='latex-bold'>(D)</span> \ 12 \qquad <span class='latex-bold'>(E)</span> \ 16
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