Geometry: An Interactive Journey to Mastery

Lesson 32

1. $Q\OLQHWKURXJKWKHFHQWHURIDUHJXODUSRO\JRQZLWKDQHYHQQXPEHURIVLGHVGLYLGHVWKHDUHDRIWKDW
   SRO\JRQH[DFWO\LQKDOIͼ௘'UDZDSLFWXUHWRVHHZK\௘ͽ7KXVWKHVWUDLJKWOLQHWKDWSDVVHVWKURXJKWKHFHQWHU
   RIWKHKH[DJRQDQGWKHFHQWHURIWKHGHFDJRQGLYLGHVWKHDUHDVRIWKHKH[DJRQDOKROHLQKDOIDQGWKHDUHD
   RIWKHHQWLUHGHFDJRQLQKDOI,WQRZIROORZVWKDWLWFXWVWKHDUHDRIWKHEURZQLHVZLWKWKHKROHPLVVLQJLQ
   half, too.


2. $Q\OLQHWKURXJKWKHFHQWHURIDUHJXODUSRO\JRQZLWKDQHYHQQXPEHURIVLGHVGLYLGHVWKHSHULPHWHURIWKDW
   SRO\JRQH[DFWO\LQKDOIͼ௘'UDZDSLFWXUHWRVHHZK\௘ͽ7KXVWKHOLQHGHVFULEHGLQ3UREOHPDFFRPSOLVKHG
   DOORIWKHIHDWVOLVWHGLQWKLVSUREOHP


3. %\LPDJLQLQJDPLUURUUHÀHFWLRQRIWKHVWULQJDQGWKH
   region it encloses, we see that we are trying to enclose a
   region along a straight line using a perimeter of 80 inches.
   )URP'LGR¶VSUREOHPZHNQRZWKDWDVHPLFLUFOHJLYHVWKH
   PD[LPXPSRVVLEOHDUHD7KXVXVLQJWKHLQFKHVRIVWULQJ
   WRFUHDWHDTXDUWHUFLUFOHJLYHVWKHPD[LPDODUHDLQWKLV
   SUREOHP௘6HHFigure S.32.1௘ͽ


4. 6XSSRVHWKDWWKHVTXDUHKDVVLGHOHQJWKx.
   'UDZWKHKHLJKWVRIWKHWZRVKDGHGWULDQJOHVFDOOLQJWKHLUOHQJWKV
   a and b, as shown in Figure S.32.2. Clearly, a + b = x.
   Then, the total area of the portion of the square shaded gray is

(^111232) xa  bx x a b 12 x (^2) , which is half the area of the square.
7KHWRWDODUHDRIWKHZKLWHSRUWLRQVPXVWEHWKHUHPDLQLQJKDOIRIWKHDUHD
of the square.

5. 1R7KHOLQHWKURXJKWKHFHQWHUSDUDOOHOWRWKHEDVH
   IRUH[DPSOHGRHVQRW௘6HHFigure S.32.3௘ͽ
   ͼ௘:KDWLVDQHOHJDQWZD\WRVHHWKDWWKHUDWLRRIWKHDUHDV
   RIWKHWZRUHJLRQVRIWKHWULDQJOHFXWE\WKLVOLQHLV"௘ͽ

Figure S.32.1

ab

x

Figure S.32.2

Figure S.32.3