Question

calcule a medida de AC dos triangulos

ED//BC
AE= 9cm
ED= 12 cm
BC= 30 cm​

Answer 1

Como ED é paralelo a BC, podemos afirmar que, na (a), os triangulos ABC e ADE são semelhantes e, na (b), os triângulos ABC e ADE são também semelhantes.

Sendo assim, podemos montar relações entre suas medidas, acompanhe.

a)

$\frac{CB}{DE}~=~\frac{AC}{AE}\\\\\\\frac{30}{12}~=~\frac{AC}{9}\\\\\\Multiplicando~Cruzado\\\\\\12~.~AC~=~30~.~9\\\\\\12AC~=~270\\\\\\AC~=~\frac{270}{12}\\\\\\\boxed{AC~=~22,5\,cm}$

b)

$\frac{BC}{DE}~=~ \frac{AB}{AD}\\\\\\\frac{36}{20}~=~\frac{12+x}{x}\\\\\\Multiplicando~Cruzado\\\\\\36~.~x~=~20~.~(12+x)\\\\\\36x~=~20x+240\\\\\\36x-20x~=~240\\\\\\16x~=~240\\\\\\x~=~\frac{240}{16}\\\\\\\boxed{x~=~15\,m}$

Aplicando Pitágoras no triangulo ABC para determinar AC:

$AC^2~=~AB^2+BC^2\\\\\\AC^2~=~(x+12)^2+36^2\\\\\\AC^2~=~(15+12)^2+1296\\\\\\AC^2~=~729+1296\\\\\\AC~=~\sqrt{2025}\\\\\\\boxed{AC~=~45\,m}$