Geometry: An Interactive Journey to Mastery

(Greg DeLong) #1

WKDWJHRPHWU\PXVWEHJLQE\LGHQWLI\LQJEHOLHIVDERXWWKHPDWKHPDWLFDOZRUOGWKDWZHIHHOVKRXOGEHDFFHSWHG


DVWUXH7KHJUHDW*UHHNJHRPHWHU(XFOLGͼ௘F%&(௘ͽLGHQWL¿HGEHJLQQLQJDVVXPSWLRQVWRDOORIJHRPHWU\
DQGVKRZHGLQKLVYROXPHWH[WThe ElementsWKDWWKHHQWLUHVXEMHFWORJLFDOO\XQIROGVIURPWKHP+LVWH[WLV
KDLOHGDVRQHRIWKHJUHDWLQWHOOHFWXDODFKLHYHPHQWVRIPDQNLQGLQWKHHQWLUHKLVWRU\RIPDQNLQG7KHSURFHVVRI
ORJLFDOGHGXFWLRQ(XFOLGGHPRQVWUDWHGGH¿QHGWKHYHU\QDWXUHRIPRGHUQVFLHQWL¿FWKRXJKW
Example 1: Egyptian Ropes
$URSHRIOHQJWKGLYLGHGLQWRVWUDLJKWVHFWLRQVRIOHQJWKVDQG
PDNHVDWULDQJOHWKDWVLWVVQXJO\LQWKHULJKWDQJOHEHWZHHQDZDOODQG
WKHÀRRUͼ௘6HHFigure 1.1.௘ͽ
([SHULPHQWZLWKDURSHRIOHQJWKLQFKHV&DQ\RXGLYLGHLWLQWRWKUHH
OHQJWKVWKDWDOVRPDNHDWULDQJOHWKDWVLWVVQXJO\EHWZHHQWKHZDOODQG
WKHÀRRU":KDWDUHWKHWKUHHOHQJWKV\RX¿QG"
Solution
7KURXJKWULDODQGHUURU\RXFDQLQGHHGXVHWKHURSHWRPDNHVXFKDWULDQJOH7KHWKUHHOHQJWKV\RXFUHDWHWKLV
ZD\ZLOOOLNHO\KDYHPHDVXUHVWKDWDUHDZNZDUGIUDFWLRQDOQXPEHUV,I\RXKDYHWKHSDWLHQFH\RXFDQYHULI\
WKDWWKHWKUHHZKROHQXPEHUOHQJWKVDQGͼ௘DGGLQJWR௘ͽZRUNQLFHO\WRPDNHDVQXJ¿WWLQJWULDQJOH
Example 2: Right Angles in Circles
7UDFHWKHSHULPHWHURIDGULQNLQJJODVVRQDSLHFHRISDSHU8VLQJDVHFRQGSLHFHRISDSHUDQGDUXOHUKRZFRXOG
\RX¿QGWKHGLDPHWHURI\RXUJODVV"
Solution
/D\WKHFRUQHURIWKHVHFRQGSLHFHRISDSHURQWKHFLUFOH\RXGUHZDV
VKRZQͼ௘6HHFigure 1.2.௘ͽ0DUNZKHUHWKHHGJHVRIWKLVSDSHULQWHUVHFW
WKHFLUFOHDQGPHDVXUHWKHGLVWDQFHEHWZHHQWKRVHWZRSRLQWV7KLVLVWKH
GLDPHWHURI\RXUGULQNLQJJODVV
Study Tip
x 7KHVWXG\RIJHRPHWU\UHTXLUHVDIUDQNZLOOLQJQHVVWRTXHVWLRQ
LGHDVDVVXPSWLRQVDQGEHOLHIV$VZHZLOOVHHWKURXJKRXWWKLV
FRXUVHLQWXLWLRQFDQOHDGXVDVWUD\'RQ¶WEHPLVOHGE\UHVXOWVWKDWRQO\seemWUXH6XFFHVVFRPHVIURP
WDNLQJWKHWLPHWRUHÀHFWRQLGHDVDQGWRTXHVWLRQDVVXPSWLRQV7KHUHLVQRQHHGWRKXUU\LQWKHVWXG\
RIJHRPHWU\


3


4


5


Figure 1.

Figure 1.
Free download pdf