Chuck Kuffner brings us this story about the new Texas Assessment of Nollige & Skilz (TAKS) test. In the math section of the test is a question that apparently cannot be answered, since it is total nonsense. Now, I'm reasonably good at math, but math does two things for me: 1) it gives me a headache, and 2) it puts me to sleep.
Chuck is even nice enough to attempt an explanation of how it's all supposed to work.
The real problem, as the various mathematicians and engineers quoted in the story note, is that the problem is impossible on its face. Here's why:
In any regular polygon, the sum of the exterior angles is always 360 degrees. For an octagon, that means each exterior angle is 45 degrees, which in turn means that each interior angle is 135 degrees.
In the triangle drawn within the octagon, the two sides that go from the center of the octagon to one of the interior angles bisects that angle. That means that the angle formed by the bottom segment of the octagon (whose length we need to calculate in order to find the perimeter) and the line whose length is given as 4.6 cm is 67.5 degrees.
(Even if you didn't know all that, if you observe that the angle from the center of the octagon to the two corners as drawn must be 45 degrees, then the other two angles in the isoceles triangle must be 67.5 degrees each.)
Now then. In a right triangle, one can calculate the sines of its angles by the formula sine = opposite/hypotenuse. (Cosine is adjacent/hypotenuse and tangent is opposite/adjacent, leading to the mnemonic SOHCAHTOA.) In the drawing, that would make the sine of 67.5 degrees about 0.870. Unfortunately, the angle whose sine is 0.87 is about 60.4 degrees. The actual sine of 67.5 degrees is about 0.923. In other words, a right triangle with a 67.5 degree angle cannot have a side measuring 4 cm and a hypotenuse measuring 4.6 cm. It's impossible.
Right; and it's so much easier when you break things down scientifically, eh?? Sorry, Chuck, but you lost me at "The real problem...." What can I say? I was a History and Anthopology major. Of course, Chuck IS a whole lot smarter than I am, so I'll take his word on this.
All of this math is aggravating my migraine; I think I need to go lie down....
