Cho lăng trụ đứng tam giác ABC.A'BC' có đáy là một tam giác vuông cân tại B, AB = BC = a,\(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiqadg % eagaqbaiabg2da9iaadggadaGcaaqaaiaaikdaaSqabaaaaa!3A4F! AA' = a\sqrt 2\) , M là trung điểm BC . Tính khoảng cách giữa hai đường thẳng AM và B'C.
Suy nghĩ trả lời câu hỏi trước khi xem đáp án
Lời giải:
Báo sai.png)
Gọi E là trung điểm của BB'. Khi đó: \(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyraiaad2 % eacaaMi8UaaGjbVlaab+cacaqGVaGaaGjbVlqadkeagaqbaiaadoea % aaa!3F39! EM{\kern 1pt} \;{\rm{//}}\;B'C\)\(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4Tabm % OqayaafaGaam4qaiaayIW7caaMe8Uaae4laiaab+cacaaMe8Uaaiik % aiaadgeacaWGnbGaamyraiaacMcaaaa!43B6! \Rightarrow B'C{\kern 1pt} \;{\rm{//}}\;(AME)\)
Ta có:\(d\left( {AM,B'C} \right) = d\left( {B'C,\left( {AME} \right)} \right) = d\left( {C,\left( {AME} \right)} \right) = d\left( {B,\left( {AME} \right)} \right)\)
Xét khối chóp BAME có các cạnh BE ,AB, BM đôi một vuông góc với nhau nên \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca % aIXaaabaGaamizamaaCaaaleqabaGaaGOmaaaakmaabmaabaGaamOq % aiaacYcadaqadaqaaiaadgeacaWGnbGaamyraaGaayjkaiaawMcaaa % GaayjkaiaawMcaaaaacqGH9aqpdaWcaaqaaiaaigdaaeaacaWGbbGa % amOqamaaCaaaleqabaGaaGOmaaaaaaGccqGHRaWkdaWcaaqaaiaaig % daaeaacaWGnbGaamOqamaaCaaaleqabaGaaGOmaaaaaaGccqGHRaWk % daWcaaqaaiaaigdaaeaacaWGfbGaamOqamaaCaaaleqabaGaaGOmaa % aaaaaaaa!4C37! \frac{1}{{{d^2}\left( {B,\left( {AME} \right)} \right)}} = \frac{1}{{A{B^2}}} + \frac{1}{{M{B^2}}} + \frac{1}{{E{B^2}}}\)\(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HS9aaS % aaaeaacaaIXaaabaGaamizamaaCaaaleqabaGaaGOmaaaakmaabmaa % baGaamOqaiaacYcadaqadaqaaiaadgeacaWGnbGaamyraaGaayjkai % aawMcaaaGaayjkaiaawMcaaaaacqGH9aqpdaWcaaqaaiaaiEdaaeaa % caWGHbWaaWbaaSqabeaacaaIYaaaaaaaaaa!4588! \Leftrightarrow \frac{1}{{{d^2}\left( {B,\left( {AME} \right)} \right)}} = \frac{7}{{{a^2}}}\)\(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSTaam % izamaaCaaaleqabaGaaGOmaaaakmaabmaabaGaamOqaiaacYcadaqa % daqaaiaadgeacaWGnbGaamyraaGaayjkaiaawMcaaaGaayjkaiaawM % caaiabg2da9maalaaabaGaamyyamaaCaaaleqabaGaaGOmaaaaaOqa % aiaaiEdaaaaaaa!44C7! \Leftrightarrow {d^2}\left( {B,\left( {AME} \right)} \right) = \frac{{{a^2}}}{7}\)\(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSTaam % izamaabmaabaGaamOqaiaacYcadaqadaqaaiaadgeacaWGnbGaamyr % aaGaayjkaiaawMcaaaGaayjkaiaawMcaaiabg2da9maalaaabaGaam % yyaaqaamaakaaabaGaaG4naaWcbeaaaaaaaa!42FC! \Leftrightarrow d\left( {B,\left( {AME} \right)} \right) = \frac{a}{{\sqrt 7 }}\)
.
