Question

In triunghiul dreptunghic ABC cu masura unghiului A egal cu 90 grade si masura unghiului B egal cu 60 de grade , [AM] este inaltimea si N este mijlocul lui [BC] .

Stiind ca MN egal 6 cm , se cere:

Masura lui AB si masura lui BC

 

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Answer 1

in ΔABM Daca <ABM=60=>
<BAM=180-<ABM-<BMA=180-60-90=30
=> <MAC=<BAC-<BAM=90-30=60

Daca <B=60 si <A=90=> <C=30
In ΔANC daca N este mijlocul lui BC atunci AN este Mediana => AN=NC
=> <NAC=<NCA=30

< MAN=MAC-<NAC=60-30=30
in ΔMAN, <AMN=90
sin <MAN=sin 30=MN/AN
MN=6
1/2=6/AN
AN=12=NC=BN (pt ca AN este mediana corespunzatoare ipotenuzei in triunghiul dreptunghic)
BC=2*NC
BC=24

BM=BN-MN=12-6=6
Avem Teorema catetei:
AB²=BM*BC=6*24=144
AB=12