Răspuns:
Explicație pas cu pas:
ΔABC, ∡A=90°. AB=12cm, AC=12√3cm. MA⊥(ABC), MA=12√3cm.
a) MA⊥(ABC),⇒MA⊥AB si MA⊥AC.
MB²=MA²+AB²=(12√3)²+12²=12²·(3+1)=12²·4. Deci MB=√(12²·4)=12·2=24cm.
MC²=MA²+AC²=2·(12√3)². Atunci MC=12√3·√2=12√6cm.
b) AD⊥BC, D∈BC. Atunci dupa T3⊥, ⇒MD⊥BC. ΔMAD dreptunghic in A
Din ΔABC, BC²=AB²+AC²=12²+(12√3)²=12²·4, deci BC=24cm
Din formula ariei, (1/2)·AB·AC=(1/2)·BC·AD |·2, ⇒12·12√3=12√3·AD, deci AD=12cm. Atunci MD²=MA²+AD²=(12√3)²+12²=24cm.
c) d(C,(MAD))=???
(MAD)⊥(ABC), deci d(C,(MAD))=d(C,AD). AD⊥BC, deci CD⊥AD si deci d(C,AD)=CD. Din teorema catetei, AC²=CD·BC, ⇒(12√3)²=CD·24, ⇒
CD=(12√3)²:24=(12·12·3):24=18cm=d(C,(MAD)).
