Thinking Outside 3 Points
A triangle on the coordinate plane has vertices , , and , where is a real number. The minimum value of the perimeter of this triangle can be written as , where and are integers. What is ?
The answer is 110.
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1 solution
Through the distance formula, the distance between the fixed points (3,15) and (10,9) is 8 5 . Reflect the point (10,9) across the x-axis to get the point (10,-9). These two points are the same distance apart from the point (x,0). Therefore, the minimum value of the sum of the lengths of the other two sides of the triangle can be calculated by adding the lengths of the line from (3,15) to (x,0) and the line from (x,0) to (10,-9). The shortest distance between these two points would be a straight line which intersects the point (7.375,0) and has the length 25 due to the distance formula. Adding this with the length of the fixed points would mean the minimum value of the perimeter is 2 5 + 8 5 and the answer is 25+85=110.