Cho hình chóp đều S.ABC có cạnh đáy bằng a , góc giữa một mặt bên và mặt đáy bằng \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOnaiaaic % dacqGHWcaSaaa!395A! 60^\circ \) . Tính độ dài đường cao SH.
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Lời giải:
Báo saiGọi M là trung điểm của BC
Vì \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiqaaqaabe % qaamaabmaabaGaam4uaiaadkeacaWGdbaacaGLOaGaayzkaaGaeyyk % IC8aaeWaaeaacaWGbbGaamOqaiaadoeaaiaawIcacaGLPaaacqGH9a % qpcaWGcbGaam4qaaqaaiaadofacaWGnbGaeyOGIW8aaeWaaeaacaWG % tbGaamOqaiaadoeaaiaawIcacaGLPaaacaGG6aGaam4uaiaad2eacq % GHLkIxcaWGcbGaam4qaaqaaiaadgeacaWGnbGaeyOGIW8aaeWaaeaa % caWGbbGaamOqaiaadoeaaiaawIcacaGLPaaacaGG6aGaamyqaiaad2 % eacqGHLkIxcaWGcbGaam4qaaaacaGL7baaaaa!5D5D! \left\{ \begin{array}{l} \left( {SBC} \right) \cap \left( {ABC} \right) = BC\\ SM \subset \left( {SBC} \right):SM \bot BC\\ AM \subset \left( {ABC} \right):AM \bot BC \end{array} \right.\).\(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H49aae % caaeaacaWGtbGaamyqaiaad2eaaiaawkWaaiabg2da9iaaiAdacaaI % WaWaaWbaaSqabeaacaqGVbaaaaaa!3F22! \Rightarrow \widehat {SAM} = {60^{\rm{o}}}\)
Gọi H là trọng tâm tam giác ABC . Vì S.ABC là hình chóp đều nên \(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaiaadI % eacqGHLkIxdaqadaqaaiaadgeacaWGcbGaam4qaaGaayjkaiaawMca % aaaa!3D27! SH \bot \left( {ABC} \right)\).
Trong tam giác SHM vuông có \(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaiaadI % eacqGH9aqpcaWGibGaamytaiaac6caciGG0bGaaiyyaiaac6gacaaI % 2aGaaGimaiabgclaWkabg2da9maalaaabaGaamyyamaakaaabaGaaG % 4maaWcbeaaaOqaaiaaiAdaaaGaaiOlamaakaaabaGaaG4maaWcbeaa % kiabg2da9maalaaabaGaamyyaaqaaiaaikdaaaGaaiOlaaaa!49C2! SH = HM.\tan 60^\circ = \frac{{a\sqrt 3 }}{6}.\sqrt 3 = \frac{a}{2}.\)