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- trapézio.png (17.73 KiB) Exibido 2292 vezes
Observe que:
[tex3]Area_{ABNM}=\frac{(a+c)d}{2}[/tex3]
[tex3]Area_{MNCD}=\frac{(c+b)e}{2}[/tex3]
E também que:
[tex3]\frac{(a+c)d}{2}=\frac{(c+b).e}{2}[/tex3]
[tex3](a+c)d=(c+b)e[/tex3]
[tex3]\boxed{\frac{e}{d}=\frac{(a+c)}{(c+b)}}[/tex3]
Perceba também que:
[tex3]Area_{ABCD}=\frac{(a+b)(d+e)}{2}[/tex3]
Portanto
[tex3]\frac{(a+b)(d+e)}{2}=2.\frac{(a+c)d}{2}[/tex3]
[tex3](a+b)(d+e)=2d(a+c)[/tex3]
[tex3]\frac{2(a+c)}{(a+b)}=\frac{(d+e)}{d}[/tex3]
[tex3]\frac{2(a+c)}{(a+b)}=1+\frac{(a+c)}{(b+c)}[/tex3]
[tex3]\frac{2(a+c)}{(a+b)}=\frac{(2c+a+b)}{(b+c)}[/tex3]
[tex3]2(a+c)(b+c)=(2c+a+b)(a+b)[/tex3]
[tex3]2ab+2ac+2cb+2c^2=2ac+a^2+ab+2cb+ab+b^2[/tex3]
[tex3]2c^2=a^2+b^2[/tex3]
[tex3]c^2=\frac{a^2+b^2}{2}[/tex3]
[tex3]\boxed{\boxed{\boxed{c=\sqrt{\frac{a^2+b^2}{2}}}}}[/tex3]
Att L.O
