Question

sa se afle ariile a două triunghiuri stiind ca raportul luminilor lor este 4/3 raportul laturilor
lor este 7/9 iar diferența ariilor este 4.
Vă rog ajutati-mă ​

Answer 1

$\it \mathcal{A}_1=\dfrac{b_1h_1}{2} \Rightarrow 2\mathcal{A}_1 =b_1h_1\\ \\ \\ \mathcal{A}_2=\dfrac{b_2h_2}{2} \Rightarrow 2\mathcal{A}_2 = b_2h_2$

$\it \dfrac{2\mathcal{A}_1}{2\mathcal{A}_2} =\dfrac{b_1h_1}{b_2h_2} \Rightarrow \dfrac{\mathcal{A}_1}{\mathcal{A}_2} =\dfrac{b_1h_1}{b_2h_2}\ \ \ \ \ (1)$

$\it \dfrac{b_1}{b_2}=\dfrac{7}{9} \ \ \ \ \ (2)\\ \\ \\ \dfrac{h_1}{h_2}=\dfrac{4}{3}\ \ \ \ \ (3) \\ \\ \\ (2),\ (3) \Rightarrow \dfrac{b_1h_1}{b_2h_2}=\dfrac{28}{27} \ \ \ \ \ \ (4)$

$\it (1),\ (4) \Rightarrow \dfrac{\mathcal{A}_1}{\mathcal{A}_2}=\dfrac{28}{27}\ \stackrel{derivare}{\Longrightarrow}\ \dfrac{\mathcal{A}_1-\mathcal{A}_2}{\mathcal{A}_2}=\dfrac{28-27}{27} \Rightarrow\\ \\ \\ \Rightarrow \dfrac{4}{\mathcal{A}_2} =\dfrac{1}{27} \Rightarrow \mathcal{A}_2 =4\cdot 27 \Rightarrow \mathcal{A}_2 =108\ cm^2$

$\it \mathcal{A}_1 -\mathcal{A}_2 =4 \Rightarrow \mathcal{A}_1 =\mathcal{A}_2+4 \Rightarrow \mathcal{A}_1=108+4 \Rightarrow \mathcal{A}_1=112\ cm^2$