Răspuns:
a) 60 cm²; b) 45°
Explicație pas cu pas:
ABCD este pătrat => AB = BC = CD = AD = 12 cm
a) AE = EB = ½×AB = 6 cm
BF = 2CF
BF + CF = BC => 3CF = BC => CF = 4 cm => BF = 8 cm
Aria(ΔDEF) = Aria(ABCD) - [Aria(ΔAED) + Aria(ΔDCF) + Aria(ΔEBF)] = AB² - (½×AD×AE + ½×CD×CF + ½×EB×BF) = 12² - (½×12×6 + ½×12×4 + ½×6×8) = 144 - (36 + 24 + 24) = 144 - 84 = 60
=> Aria(ΔDEF) = 60 cm²
b) DE² = AD² + AE² = 12² + 6² = 144 + 36 = 180
=> DE = 6√5 cm
DF² = CD² + CF² = 12² + 4² = 144 + 16 = 160
=> DF = 4√10
[tex]Aria_{\triangle DEF} = \frac{DE \cdot DF \cdot \sin(EDF)}{2} \\ \frac{6 \sqrt{5} \cdot 4 \sqrt{10} \cdot \sin(EDF)}{2} = 60 \\ \sin(\angle EDF) = \frac{1}{ \sqrt{2} } \implies \bf \angle EDF = 45 \degree[/tex]
q.e.d.
