Ensino Fundamental ⇒ Triângulos retângulos

[Marycs09]

Na figura abaixo, ABCD e EFGH são quadrados com lados 7 cm e 5 cm, respectivamente. Sabendo que EQ < QF, determine o perímetro da região destacada.

[petrasMOD]

@Marycs09,
[tex3]S_{ABCD} - S_{EFGH} = 7^2-5^2 = 24 \implies S_{AEF} = S_{FBG}=S_{CGH}=S_{HDE} = \frac{24}{4} = 6\\
\triangle AEF: S_{AEF} = \frac{EF.AQ}{2} = \frac{5.h}{2}=6 \implies AQ = \frac{12}{5}=PD=SC=RB\\
\triangle AEF: AQ^2=EQ.QF \implies (\frac{12}{5})^2 = x(5-x) \implies \frac{144}{25} = 5x - x^2 \\
25x^2-125x+144=0 \implies \cancel{x = \frac{16}{5}}:x' = \frac{9}{5}(x < 5- x \implies x < \frac{5}{2} )\\
\triangle AQE: AE^2 = x^2+AQ^2 = (\frac{9}{5})^2+(\frac{12}{5})^2 = \frac{225}{25} = \frac{45}{5} = 9\\
\therefore AE = 3\\
2p = 4AE+4EP+4PD = 4.3+4.(5-\frac{9}{5})+4.\frac{12}{5} = 12+\frac{64}{5}+\frac{48}{5} = \frac{172}{5} = \boxed{34,4}[/tex3]

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