1957 AHSME Problems/Problem 50
Problem
In circle , is a moving point on diameter . is drawn perpendicular to and equal to . is drawn perpendicular to , on the same side of diameter as , and equal to . Let be the midpoint of . Then, as moves from to , point :
Solution
Let and . Then, we know that the diameter of the circle equals . Thus, because the circle's diameter does not change, is constant.
Because and , . Thus, , and so is the distance from to .
Let be some point which moves along . Because is a line segment, as moves from to , its distance from will vary linearly with how much it has travelled along . Thus, when it is halfway along (in other words, when ) its distance from will be the arithmetic mean of its distance from at (namely, ) and its distance from at (namely, ). Thus, .
Because , a constant, is a constant as well. Thus, is the same regardless of the position of . Furthermore, from our work in paragraph 2, we know that must lie on the line perpendicular to through point . Therefore, because is a fixed distance from a fixed point on a fixed line, and it will not suddenly "jump across" to the other side of , we can say with confidence that point .
See Also
1957 AHSC (Problems • Answer Key • Resources) | ||
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