{"id":467,"date":"2019-06-29T21:53:27","date_gmt":"2019-06-29T21:53:27","guid":{"rendered":"http:\/\/funfacts.104.42.120.246.xip.io\/?page_id=467"},"modified":"2019-12-20T23:22:06","modified_gmt":"2019-12-20T23:22:06","slug":"sliding-chords","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/funfacts\/sliding-chords\/","title":{"rendered":"Sliding Chords"},"content":{"rendered":"\n
\"\"<\/figure><\/div>\n\n\n\n

Take a circle C, and a\u00a0chord\u00a0in the circle. Now slide the chord around the circle. As you do this, the midpoint of the curve will trace out a smaller concentric circle. Call the area between the two circles A(C).<\/p>\n\n\n\n

Now suppose you do the same thing with a larger circle C’ but with the same length chord? Will A(C’) be larger or smaller than A(C)?<\/p>\n\n\n\n

Surprise: they will actually be the same area! In otherwords A(C) does not depend on what circle C you start with, only the length of the chord!<\/p>\n\n\n

Presentation\u00a0Suggestions:<\/strong>
From this fact, ask students if they can see quickly what the fixed area must be! [Hint: start with a circle whose diameter is the length of the chord.]<\/p>\n

The\u00a0Math\u00a0Behind\u00a0the\u00a0Fact:<\/strong>
In fact, an even more amazing fact is true: take\u00a0any\u00a0convex\u00a0shape C<\/em>\u00a0and place a chord of fixed length in it. Now slide as you slide the chord around C, the midpoint traces out another figure D. The area between C and D does not depend on what shape you started with!<\/p>\n

How to Cite this Page:<\/strong>\u00a0
Su, Francis E., et al. “Sliding Chords.”\u00a0Math Fun Facts<\/em>. <http:\/\/www.math.hmc.edu\/funfacts>.<\/p>\n\n\n

References:<\/strong>
R. Vakil, A Mathematical Mosaic<\/a>, 1996.<\/p>\n\n\n\n

Fun Fact suggested by: <\/strong>
Ravi Vakil <\/p>\n","protected":false},"excerpt":{"rendered":"

Take a circle C, and a\u00a0chord\u00a0in the circle. Now slide the chord around the circle. As you do this, the...<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[20,7,8,4],"class_list":["post-467","page","type-page","status-publish","hentry","tag-analysis","tag-calculus","tag-geometry","tag-medium"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/467","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/comments?post=467"}],"version-history":[{"count":4,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/467\/revisions"}],"predecessor-version":[{"id":1629,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/467\/revisions\/1629"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=467"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=467"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}