{"id":5636,"date":"2013-12-29T20:45:49","date_gmt":"2013-12-30T01:45:49","guid":{"rendered":"http:\/\/spinor.info\/weblog\/?p=5636"},"modified":"2014-01-08T14:56:14","modified_gmt":"2014-01-08T19:56:14","slug":"pointless-mathematics","status":"publish","type":"post","link":"https:\/\/spinor.info\/weblog\/?p=5636","title":{"rendered":"Pointless mathematics"},"content":{"rendered":"

The other day, I ran across a cute geometry puzzle<\/a> on John Baez’s Google+ page. I was able to solve it in a few minutes, before I read the full post that suggested that this was, after all, a harder-than-usual area puzzle. Glad to see that, even though the last high school mathematics competition in which I participated was something like 35 years ago, I have not yet lost the skill.<\/p>\n

Anyhow, the puzzle is this: prove that the area of the two semicircles below is exactly half the area of the full circle.<\/p>\n

\"\"<\/p>\n

I am going to insert a few blank lines here before providing my solution.<\/p>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n

I start with labeling some vertices on the diagram and also drawing a few radii and other lines to help.<\/p>\n

\"\"<\/p>\n

Next, let’s call the radii of the two semicircles as \\(a\\) and \\(b\\). Then, we have
\n\\begin{align}
\n(AC)&= a,\\\\
\n(BD)&= b.
\n\\end{align}Now observe that
\n\\begin{align}
\n(OA) = (OB) = r,
\n\\end{align}and also
\n\\begin{align}
\n(CD)&= a + b,\\\\
\n(OD)&= a + b~- (OC).
\n\\end{align}The rest is just repeated application of the theorem of Pythagoras:
\n\\begin{align}
\n(OC)^2&= r^2 – a^2,\\\\
\n(OD)^2&= r^2 – b^2,
\n\\end{align}followed by a bit of trivial algebra:
\n\\begin{align}
\n(OC)^2 + a^2&= [a + b – (OC)]^2 + b^2,\\\\
\n0&= 2(a + b)[b – (OC)],\\\\
\n(OC)&= b.
\n\\end{align}Therefore,
\n\\begin{align}
\na^2+b^2=r^2,
\n\\end{align}which means that the area of the full circle is twice the sum of the areas of the two semicircles, which is what we set out to prove.<\/p>\n

I guess I have not yet lost my passion for pointless, self-serving mathematics.<\/p>\n<\/fb:like>","protected":false},"excerpt":{"rendered":"

The other day, I ran across a cute geometry puzzle on John Baez’s Google+ page. I was able to solve it in a few minutes, before I read the full post that suggested that this was, after all, a harder-than-usual area puzzle. Glad to see that, even though the last high school mathematics competition in […]<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[30],"tags":[],"class_list":["post-5636","post","type-post","status-publish","format-standard","hentry","category-mathematics","category-30-id","post-seq-1","post-parity-odd","meta-position-corners","fix"],"_links":{"self":[{"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=\/wp\/v2\/posts\/5636","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=5636"}],"version-history":[{"count":9,"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=\/wp\/v2\/posts\/5636\/revisions"}],"predecessor-version":[{"id":5696,"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=\/wp\/v2\/posts\/5636\/revisions\/5696"}],"wp:attachment":[{"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5636"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5636"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5636"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}