Question

rombul abcd are aria 24√3cm pătrați si masura unghiului A de 60° Lungimea diag ac este de...
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Answer 1

Răspuns:

Explicație pas cu pas:

Daca ∡A=60°, atunci ΔABD echilateral, deci diagonala BD=AB.

Aria(ΔABD)=(1/2)·Aria(romb)=(1/2)·24√3=12√3cm².

Dar Aria(ΔABD)=AB²·√3/4. Deci AB²·√3/4=12√3. Deci AB²=4·12=4²·3

Deci AB=4√3cm=BD.

Aria(romb)=(1/2)·BD·AC=24√3, ⇒(1/2)·4√3·AC=24√3, ⇒AC=12cm

Answer 2

$\it Fie\ \ell\ latura\ rombului.\\ \\ \Delta ABD-isoscel\ cu\ m(\hat{A})=60^o \Rightarrow \Delta ABD-echilateral \Rightarrow\\ \\ \Rightarrow AB=BD=AD=\ell\\ \\ \mathcal{A}_{ABD}=\dfrac{\mathcal{A}_{ABCD}}{2}\Rightarrow \dfrac{\ell^2\sqrt3}{4}=\dfrac{^{2)}24\sqrt3}{2}\Rightarrow \dfrac{\ell^2\sqrt3}{4}=\dfrac{48\sqrt3}{4}\Rightarrow \ell^2\sqrt3=48\sqrt3$

$\it \Rightarrow \ell^2=48 \Rightarrow \ell=\sqrt{48}=\sqrt{16\cdot3}=4\sqrt3cm \Rightarrow BD=4\sqrt3\ cm\\ \\ \mathcal{A}_{ABCD} =\dfrac{BD\cdot AC}{2}=24\sqrt3 \Rightarrow \dfrac{4\sqrt3\cdot AC}{2}=24\sqrt3 \Rightarrow2\sqrt3\cdot AC=24\sqrt3|_{:2\sqrt3}\Rightarrow\\ \\ \\ \Rightarrow AC=12\ cm$