Geometry: An Interactive Journey to Mastery



Solutions

8. D௘ͽ $UHD î 

E௘ͽ ʌU JLYHVr (^20) S, so area SSr^2  S (^2) S.
F௘ͽ   rr 21 S That is, 2r + ʌU ZKLFKJLYHVr 2 S.
So, area  222 Sr^2 SS§·©¹ ̈ ̧S^2 2 S 2.

9. D௘ͽ 7KHLUDUHDVKDYHWKHVDPHQXPHULFDOYDOXHDVWKHLUSHULPHWHUV
   E௘ͽ 7KHWULDQJOHKDVSHULPHWHUDQGDUHD
   F௘ͽ 1R,Ia × b = 2a + 2b, then b aa  aa 22  2.a 2
   For this to be an integer, we need aíWREHDIDFWRURI
   If aí WKHQa = 3 and b = 6, and we have the 3 × 6 rectangle.
   If aí WKHQa DQGb DQGZHKDYHWKHîVTXDUH
   If aí WKHQ a = 6 and b = 3, and we have the 3 × 6 rectangle again.
   These are the only possibilities.


10. D௘ͽ :HKDYHDIXOOWXUQRIƒGLYLGHGLQWRFRQJUXHQWSDUWV
    E௘ͽ 6$6KROGVVLGHVr and r with 30° between them and sides kr and kr with 30° between them.
    F௘ͽ %HFDXVHWKHVFDOHIDFWRULVkWKHEDVHRIHDFKWULDQJOHLQWKHODUJH¿JXUHLVk times the length of each
    EDVHLQWKHVPDOO¿JXUH6XPPLQJWKHPVKRZVWKDWWKHSHULPHWHUVDOVRGLIIHUE\DIDFWRURIk.
    G௘ͽ ,ILQWKHVPDOO¿JXUHHDFKEDVHLVx, then this ratio is^122 rx.
     )RUWKHODUJH¿JXUHWKHPDWFKLQJUDWLRLV^122 krkx^122 rx, which is the same.
    H௘ͽ 7KHUHLVQRWKLQJVSHFLDODERXWWKHQXPEHUKHUH7KHSHULPHWHUWRWZLFHUDGLXVUDWLRZRXOGDOZD\V
    PDWFKLQWKHVPDOODQGODUJH¿JXUHV