Problems 11i Geolnetr, A KUTEPOV and A RUBANOV MIR PUBLISHERS MOSCOW The book contains a collection of 1351 problems (with answers) in plane and solid geometry for technical schools and colleges The problems are of varied content, involving calculations, proof, construction of diagrams, and determination of the spatial location of geometrical points It gives sufficient problems to meet the needs of students for practical work in geometry, and the requirements of the teacher for varied material for tests, etc A It HYTEIIOB, A T PYBAHOB 3AAALIHMP no TEOMETPYIYI I'Iaj{8TeJIbcTBo 3, (2) H = R2R2 , R > 3, (3) H = 3, R-any 998 cm 999 0.26 V 1000 1:26 or 8:19 1001 24 1002 1003 240° 1004 (1) Las (2) 4800n dms, ' 13 s-j/ 2L 8i (3) 320n cros, (4) 1440n cros 1007.3525L cros 1008 3064L cm3 1009 (1) 9600a 29 s cros, (2) dms ANSWERS 201 n (26 - 15 1/3) 1010 1011 144n cm3 and 192n cm3 1012 1014 1015 2400n cm3, 960n cm2 1016 28.6 dm3 1017 72 63.3 1018 14.3 1019 1583 cm3 1020 24 cm, 25 cm 1021 22.95n 1022 24 n 1ls 1023 804.3 cm2 1024 372I3 1026. 2325cm3.1027.19:37 :61.1028 1025 1029 7na6 6na2, r 7nas 1/ 1034 1- (Rll3 27 "V3-nas 1032 7R2 + 4Rr + r2 R2+4Rr+7r2 n2R2r 1036 H = cm, R = 3.5 cm 1037 na34 1038 1039 4na2 V-2 1040 zt 1217.5 dm3 1041 6na21, 4.5na3 1042 3na3 1051 : 1052 1049 cm 1050 3585 km 15,920 km 15.7 cm 1054 2.44 radians, 1053 101.5 cm 1055 !n-R2 1056 r 48 (2 + 33 2) nR3 1030 R = 1.5 r 1031 R =r (v2+1) nRrH 1033 1035 5n1 n222 1057 30° 1058 12 dm 1059 12 cm 1060 33.8 cm 1061 75.4 cm 1062 nR x/3.2. 75.4 cm 1065 L , 1072 1066 330 m2 1067 2, 14.23 dm 183.2 cm2 1077 1073 576n cm2 1074 N 867.8 dme 1076 w3456 cm2 and 1257 cm2 1078 2500n cm2 1079 0.25 1080 26 1081 42 (5-2 -V2-) 21nQ 1082 400 (4-'l/2) n 4n-3-V-3 1088 R (1-1) 1091 s 1083 15 cm 1086 12 cm2 1092 (2) Will be reduced to 64 36nV2 27 1093 times, times , 484 times, 10,648 times 1094 1,362,385 thousand km3 1096 Yes 1097 0.3 cm 1098 (2) n, 236 n, 288n 1099 (1) (2) 4.5 1100 ,:s 1.14 1101 12 cm 1102 72 per cent 1103 cent, 55.5 per cent, 47.6 per cent 297.2 cm3 1106 2.9 cm 1107 2827 cm3 1109 1104 V-3) n 512 (16 33.3 per 4.6 cm 1105 cros 1108 416n 37 532n 11,968a cm3 cros 1(10 cm3 or '3 PROBLEMS IN GEOMETRY 202 1116 1114 18,432n cma 1115 S2 (6nRa-S) 1111 44 1113 1117 16 8000n cma 1118 : 1206 cm2 1121 24n2R3 s n3 1122 0.2 1123 V21 - 1124 If a circle can be circumscribed about base 1125 If the base is a rhombus and the distances between opposite faces are equal to each other 1126 If a circle can be circumscribed about the base of the pyramid 1127 Pos- V3 2R V3 sible If it is an equilateral one 1128 and 1/2-4) 1130 2304 dm2 1131 8R3 V2 and 2R 1129 2nR2 (3 1132 : 7794 dma 1133 (1) 18R2 V3, GRa V3, (2) 24R2, 8R3, (3) 12R2 V3, 4R3 1134 11/3 a, (3) (2) a 1138 (1) /b+ hh2+ 1/b2 - h 1139 a 1143 V6 6 , a 1144 a, (2) a (V5-1) Vb2-h2 2h 1014 cma 1136 (a) , a(y12-1) (b) , (c) 1137 (1) a (121-3) 1135 b2 2h 2- a V31x + bz + h (3) m 1145 b2- h2 ' 2h (b2-h2) 3b2 + h2 + V3 (b2 -h2) ' 1141 16 2 1/3 cm 1142 na2 (2- V3) 5236 cm3 1147 ,:s 6D6 dms IM W2 1150 nat 1151 104R2 1152 10 m 1153 (1) 1148 32 2 V2 (2)3, 3.1154 3nR3 1155 nR2 (3+2 V3) 3nR3 4 ' 3' 1156 2.25 and n12 521157 nls 1/3 4n12- W3 V 54 27 3' 12-1:2(1 254.5 m2 ^254.5 ma 1159 1160 1162 24 cm 1163 12-1t2) and 12 ' h 1158 2h 1164 900n cm2, 4500n cm3 43 169 18 dm 72 1165 15 m 1166 1064n cm2, 4256 n cma 1167 1170 h = R-a hemisphere 1171 (9-51/)8(2+ 1/2) 1172 H sin (a -}- S) sin ( - a) H cot P 1173 a2 1174 are cos a sin a sin cos (tan 2) 1175 65°25' 1177 V3 d cos a 1178 ANSWERS 203 V2Q sin (30'+2-) sin (30'-' ) 1179 1180 69°18' sin a sin (30` {- a) sin (a - 30°) 2H 54°44' 1181 2a2 sin a 1182 ° tan a cot 180) , 42°46' 1184 arctan 1183 arccos (tan cos 1185 arctan (tan a cos8) , 60° n tan ( sin a i 1188 arccos In P n 1186 70°32' M aresin 180 n 1187 arc- 1/22 sin a , 35°6' 1190 2h cot a sin 1191 arccos 1192 aresin (1sin ) 7/ 1193 2a2 27 cos a 1194 6V /2 cot a N 16.2 dm2 1196 3a2 cot a _/3 a2 sin 2a sin a 1197 16 b2 cos a 1198 HZ tan a sin a' or d sin a 1203 cost 1201 2d2 sin 2a 1202 cos a a a cos 1204 (a - b) cos a si a 157.2 cm2 1199 a2 cos3 a 1200 b tan 1195 tan tQ 1205 d cos 1206 _- 40°54' 1207 arctan oa R- d2 59°2' 1208 It sin Z m2 sin 2a tan a 1209 1210 4d2 tan2 /COs a 6030 cm2 2Q cos a, 1/Q cos a tan a 1212 l/d2 sing a-)- R2 cos2 a L2 sin (a i6n20°a sin (a - 60°) 1213 1214 1215 coss a ' 1211 H2 sin 75°31' 1216 cost (45° R2 sin a 1/sin sin ( (cos2 a s2 4) 1217 sin - a) JQ cot 1219 arccos ( cot 1218 tan 2) , a 54°44' 1220 2n sin -a 1221 aresin -SL 1222 sin -a sin 2n (R2 - r2) sin a 2Q cos P 2Q cos r aresin (R2-r2) 1223: 1224 R arcsin 2sinP (R2-t2)' PROBLEMS IN GEOMETRY 204 (a2 b2) cot a 1225 S stn 2a 1226 -h2 cot (a + P) 1227 3a Qcos2 1228 a tan 2 Q Y sins (45° - 4) 22Rr cos 2a 1232 V Ra -} si tan a 1233 116°50' 1234 2sin a+tan a 2cx (a < 120°); 1, 1, , no solution At a = 120° the conical surface is not intersected by the plane 1235 cos 2a 40 a3 , cos 2a 1236 ab aa-f- bs X sin a sin a ' X tan a 1237 d3 sin 2a cos a sin 2p 1238 d3 sin a sin X X /cos (a + P) cos (a-0) 1239 d2 sin 2a-{-2 -V-2Q cos (a-45°) Qd sin 2a.1240 a v- 2a acos a , sin 1154 cm2 1241 sin 2 d3 cot 1242 aQ sin a sin 0, 1677 cm3 X sin 2a cos a sin P 1244 U2 tan a cos 2a cos a X sin 2a sin a 1246 2955 cros 1248 d9 1/3 sin8 2a cos a 1245 d3X d3 X 1249 1250 8sin a 1.5d2 sin 2a, 2.832dms, 11.95 dma I/sin (30°+ 2) sin (30°- 2) 2a3 sin a sin 1243 Hs tans a sin 20 1247 Qd sin 2c&, asl/3 sin (30°{-2) sin (30° -x ) X d2 (2+cos a ) a2 0.6431 ms 1253 3d3 y3 sin 2a cos a 16 1251 12d2 sin 1088 dms X 1252 Q sin a YV3 Q cos a ANSWERS 1254 205 12Q sin a, 2Q sin a 1/Q cos a 1255 , sin a sin ds I,_ X sin 2(z 65.19 9.754 dms, sin a d2 Y sin 2fl cos sin 2p cos 0, 1258 dm2 cos cos 2Q 4) (45° cos (45° c32 sin2 2a 1259 pstans 45 a ) tan a tan o.12so 16 cos a cos 582.9 de as sin a tan p, 1261 X v sin 23.5 sin Us i sin 2a cos a 1265 dms 1262 12 gas -sin X 1267 2a3 sin2 a cos 1268 32 sins (45° + ) R2 sin 2a; 2R3 sin 2a cos a bs sin 2a cos a 1270 1266 a3 1269 2-sin a 1.5b2 sin 2a 1271 0.5m3 i sin 2a cos a 1272 1264 sin 32 sin a basin bQ sin a 1263 2) - 2) ((.1-45- a a3 cot cos ds X 1257 1256 1.5d2 sin a, a 1273 3821 cos$ 1274 2rs i cot a 1275 1/2 b2 sin cos a X (45°- 2) 1276 4m2 cos a (cos has 4Hs sin a 1278 V= cos a 1279 ps64tan 1280 a X sin 2a tan tan 31/(2-+ nag a)3 ' S= m3n X 1/Q cos p 1282 61/2 cos a 1283 , Z 1281 a 16R2 cos4 - V3V jr2 tan a (2 + tanla) cos a 2m2n cos a cos2 tan 180° cos a 1277 n2 Q tan 180 n cosX -/1 + sine a) 32R3 cos° a2 sine a cos ' sine a cos a 1284 PROBLEMS IN GEOMETRY 206 9p3 0093 a 13 1285 sin (60° +.a) sin (60° - 2) sin cot a cot 1286.26 b8 sin 2a sin 20 cos P 1287 as cos a cot a tan P 1288 R8 cot ( 451- 2) cot tan P 1289 213 sins a cot P Vsia+2P) sin (n-a) Tsin (2 X b8 sin X 1290 s + 60°) sin (60° -a) 1291 cos a.1294 a2 sin 1296 cot (45° - 2) 1295 X ds cot (45° - ) sine sin a 25.39 dm8 1297 COaa , a 61 c3 sin d2 tan cosy as toss cos a 1292 b8 sins y cosy sin a sin P sin (a + P) 1293 X A b2 cot 1298 24 sin 2 H (b - H s) (c - H2) sin a U.N m8 tan a sins ((x 0) sins cage a Us sins (45°+a) cot (45°-2) 1301 H8 tan cp sin 41p 1300 sins a 1302 12 (a8 - b3) tan a 1303 Bas cos2 (45° (as 1304 - b8) -cos 2w X Q (2 1/2 -1) sina nd3 sin 2a cos a, Q cos a 24 finds sin (15°-{ Q tan a, -~1132 dm8.1309 ads sin cos3 1307 sine a C3 sin 2a tan a 1310 186 dm2 1306 X 1308 80.32 cm3, ads sin 2a, 99.26 cm2 2.0.25nQ X X 1305 cos c 2) , 12 VR UP sins 2a' )cos coss2 1103 dm3 1311 (15°- ) 207' ANSWERS (a nasl/ 1312 na3 sin3 a ape sin a 4) , (2) arcsin S ad3 cot2 1319 , no cot a 450- a 2nQ cost 1316 a 2 cos a 100 cm3.1320 , na3 cos a, QJ/2nQ sin 2nd3 1322 24a sin" cos3 45 a ) 780 70 dm 1323 sin 2a sin a cos2 43X2-a2 a3a2 1324 nd3 sin2 sine - 2i 8R2 1328 sin 2a sin 450+-E 2) 1329 r2 cot cot (45° - 2) X sin 2a cos a 1333 2n12 sin cos cos a nm3 1332 (*+ 15°) cos (2 - 15°) n12 sin a tan a 1335 24 n13 sin 2a cos a 1327 12 sin Z sin a sin 1331 2) ) 2nrn3 cost a (3 sin a + coc2 cot a 2nd2 cot 1330 1325 4nb2 sin 2a cos (30°=, 2) cos (30° nb3 sine 2a 1326 n j/2 c2 sin 2a cos (45°-a) 192n2 sin3 sine a 14.32dm3 12 sin 1321 13X3 2a 6525 W nr3 cot3 tan a, Q sin-' nc3 1315 3587 CO sin a 1318 Q Q cot a nQ cost (45° 1317 n13 sin a sin , 83.83 cm3 1314 (1) 16 1/2 cos3 (a-45°) 2n12 sin cost (450 sine sin 1313 tan a 2R3 sin p toss sin (2 1336 0.5n12 3X13 X 1334 (1 + PROBLEMS IN GEOMETRY 208 32) 3a ' 6a13 tan ( cosy -COS 008 +cost 1337 stag sin a cos (30° +2) cos (30° - 2) 1338 ab3 X -} sine a- tang 2) 2 nb3 sins 2a 1339 X sin 2a cos a, 2na3 sin a sin 1, 8na2 sin , 2na3 sin a cost , 8na2 cost 1340.4 tR3 X Xsin (2+P) sin X (2-+P) sin nag tan2 2, 4nR2sin cost , 1344 (- +S) sin 1341 8nR2 sinX 445.4 cm2 1342 a2 1345 sine 4na3 1/5 nR3 sin 2a cos a, cos a 200.3 dm3 1346 nrg cots (45° - 4) cot nN2 cos a ; X sine a cost 1348 ,0S4 X Q Y sing 2a tan3 93.69 cm2 1351 1343 27 sing 2a 1347 1545 cm2 1349 3R3 X s3 -Q x 1350 Q sin a cos 4nr2 n (7 + cos 2a) Sing a' sing a cost ( 45°- ) , 1352 2nR2 sin X X(a - P)cosa+0 Mir Publishers welcome your comments on the content, translation and design of the book We would also be pleased to receive any proposals you care to make about our future publications Our address is: USSR, 129820, Moscow I-110, GSP Pervy Rizhsky Pereulok, Mir Publishers Printed in the Union of Soviet Socialist Republics OTHh'.R EMIR PCLILISHLRS' 130OKS FOR YOUR LIBRARY DescriptHy' Geometry by N KRYio%, U Sc., P LORANOIYV:V SKY, Cand, Sc., S NIF.N Geometry by P ANDREEV, Cand, Sc and E SIIUF.ALOVA, Cand Sc Problems in Elementary Mathematics for Home Study Iry N ANTONOV'; M VYCODSKY, D NIKI'I'IN, and A SANKIN Course of Mathematical Analvsis by S NIKOLSKY ... lines lies in the plane containing these lines PROBLEMS IN GEOMETRY 44 274 Straight lines a and b intersect at point M Where all straight lines lie which intersect each of the given lines and (a)... straight lines intersecting a straight line a and passing through the point A not lying on the straight line a lie in one plane Prove that a straight line intersecting two parallel straight lines... III STRAIGHT LINES AND PLANES IN SPACE Basic Concepts and Axioms Two Straight Lines in Space Straight Lines Perpendicular and Inclined to a Plane Angles Formed by a Straight Line and a Plane