Question

ABCD - paralelogram, AB = 5, AD =3, A = pi/3;
Aflati lungimea diagonalei AC.

Answer 1

Aria ABCD=AB×AD·sin∡A
AriaABCD=5·3·sin60°=15·√3/2
DUCEM CE ⊥ AB
AriaABCD=AB×CE
⇒15√3/2=5×CE ⇒CE=3√3/2
In Δ dreptunghicCEB⇒teor.Pitagora ⇒BE²+CE²=BC²
⇒BE²+27/4=9 ⇒BE²=(36-27)4 ⇒BE²=9/4⇒BE=3/2
in Δ dreptunghic AEC ⇒Teor.Pitag ⇒AE²+CE²=AC²
(13/2)²+27/4=AC² ⇒AC²=(169+27)4⇒AC²=196/4⇒ AC=14/2 ⇒AC=7cm

Answer 2

[tex]\it m(\hat{B}) = \pi - \dfrac{\pi}{3} = \dfrac{2\pi}{3},\ \ \ BC = AD = 3 \\ \\ AC^2 = AB^2+BC^2 -2\cdot AB\cdot BC\cdot cosB = 5^2+3^2-2\cdot5\cdot3\cdot cos\dfrac{2\pi}{3}[/tex]

$\it AC^2 = 25+9-30\cdot \left(-\dfrac{1}{2}\right) = 34 +15 =49 \Rightarrow AC=\sqrt{49} =7$