Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.1 AIME Find the answer United States

Problem:
A parallelogram has 3 of its vertices at $(1,2)$ , $(3,8)$ , and $(4,1)$ . Compute the sum of all possible $x$ coordinates of the 4th vertex.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:
Answer: $8$ . There are three possibilities: the 4th vertex must be opposite one of the three given vertices. These three possibilities have as a medial triangle the three given vertices, so the sum of their $x$ coordinates is the same as the sum of the $x$ coordinates of the given triangle.

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