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'Eight Ratios' printed from https://nrich.maths.org/
This problem was originally created within a theme of trigonometry.
Trigonometry can sometimes become routinised and leave students
without a good feel for trig values as ratios. This problem is
deliberately couched in ratio terms rather than Sine or Cosine
expressions. Connecting up again with Sine and Cosine could be very
valuable, especially if the extension at the end of these notes has
been explored.
Increasing the scope of this enquiry, it may be useful to ask
students whether the problem can be solved when any three of the
eight ratios are given, or must it be this particular three ?
As a very enriching extension, the eight ratios can be grouped to
produce a surprising result :
Make a group of four ratio values starting with 0.43, 0.88, and
0.62, then continue clock-wise to include the unknown ratio value
which compares the left portion of the horizontal line with the
hypotenuse of the green triangle. Form the product of these four
ratios and compare that value with the product of the other four
ratios.