Geometry: An Interactive Journey to Mastery



Solutions

Lesson 22

1. Arc length ˜ 360315 2.SSr^72 This gives^7782 ˜ 2,r so r = 2.


2. We have area 360 xxS2,^2  360 S and arc length ˜ 360 xx 22 S  360 S.
   They are the same numerical value.


3. D௘ͽ $UHD DUHDVHFWRUíDUHDWULDQJOH u ^11 SS 62 6 6 9 18.
   E௘ͽ $UHDRIVKDGHGUHJLRQ u  ^313131 area circle 1 1 SS 12.
   ௘6HHFigure S.22.1௘ͽ
   F௘ͽ %\7KDOHV¶VWKHRUHPWKHWULDQJOHKDVDULJKWDQJOH,WKDVDƒDQJOHDQG
   therefore, half an equilateral triangle.
   Its sides are 5, 10, and 53, and its area is^122 ˜u 553 25 3.
   The shaded region is half the circle minus this triangle and, therefore,
   has area^1252222 S5.^2  25 3 S 25 3
   G௘ͽ )LUVWQRWLFHWKDWIRUDFLUFOHRIUDGLXVr, the area of the shaded wedge
   shown in Figure S.22.2 is given as follows.
   $UHDVKDGHG DUHDVHFWRUíDUHDWULDQJOH
   2
   2
   2
   22

360 12 2

360

360 sin 2 cos 2

360 sin 2 cos 2.

x rbh

x rbh

xxxrr r

xxxrr

S

S

S

S

˜˜



˜§· §· ̈ ̧ ̈ ̧©¹ ©¹

§· §· ̈ ̧ ̈ ̧©¹ ©¹

90°

11

Figure S.22.1

x°

b hb

Figure S.22.2