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Two Geometrical Think-a-Lots


For these two problems, stop and think before rushing into possible solution attempts.

[1] A flexible cable of length twelve feet is hanging from two points at the same height. If the dip in the cable is six feet, determine the span.

[2] Given a triangle ABC. Is there a point X not in the plane of ABC such that AX, BX, and CX are mutually perpendicular?

Again, stop and think...a lot.

 

Source: Mathematics Magazine, 1955, p. 172 and 1957, pp. 290-291.


Hint: Try to visualize the two situations...what must be true?

 


Solution Commentary: [Solution to Problem 1] The span between the two points must be zero, as the cable must hang straight down (i.e. 6 feet down plus six feet up equals twelve feet).

[Solution to Problem 2] Think of a plane intersecting the corner of a room. The point X is the room's corner (the intersection point of three mutually intersecting perpendiculars). Now, does this work for all triangles ABC? What if ABC is a right triangle? An acute triangle? An obtuse triangle?