Why do this
|Doing this problem you need to visualise the relationship
between the formulae for the arithmetic, geometric and harmonic
means in terms of the radius of a circle and the geometry of
similar triangles and Pythagoras Theorem. This process involves a
beautiful blend of algebra and geometry. It provides a good
exercise for learners in mathematical reasoning and proof. The
mathematical concepts are simple and the interplay of ideas to give
the necessary proofs is good experience for learners.
Ask the learners to work out the radius of the semicircle in
terms of $a$ and $b$. Then ask them to identify all the triangles
and to discuss with a partner, and write down, everything they
observe about the triangles. They may then be able to do the
problem for themselves. If not the following questions should
provide some assistance.
Can you find the radius of the semicircle?
Can you see a right-angled triangle with $G$ as the length of
one side and use this to find $G$ in terms of $a$ and $b$?
Can you use similar triangles to find a relationship between
$A$, $G$ and $H$?
Can you use similar triangles to find a formula for $H$ in
terms of $a$ and $b$?
Can you you use the geometry of triangles to compare the
lengths of $A$, $G$ and $H$?
Can you see a right-angled triangle with $Q$ as the length of
one side and use this to find $Q$ in terms of $a$ and $b$?