IME / ITA ⇒ (AIME-97) Círculos tangentes Tópico resolvido

[papirador]

Os círculos de raio 5, 5, 8 e k são mutuamente tangentes externamente. Determine o valor de k.

A) 9/8
B) 8/9
C) 4/7
D) 7/4
E) 3/4

Resposta

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B

[geobson]

papirador, segue…..

[petras]

papirador,
r = ?
[tex3]x = OD\\
\triangle AOD: 5^2+x^2 = (5+r)^2 \implies x = \sqrt{10r+r^2}(I)\\
\triangle ABD: (8+r+x)^2+5^2 = 13^2 \implies(8+r+\sqrt{10r+r^2})^2=144\\
8+r+\sqrt{10r+r^2} = 12 \implies \sqrt{10r+r^2} = 4-r \implies 10r+r^2 = 16-8r+r^2\\
18r = 16 \therefore \boxed{r = \frac{8}{9}}





[/tex3]

Código: Selecionar tudoSHIFT+Click na eq = ZOOM

[tex3]x = OD\\
\triangle AOD: 5^2+x^2 = (5+r)^2 \implies x = \sqrt{10r+r^2}(I)\\
\triangle ABD: (8+r+x)^2+5^2 = 13^2 \implies(8+r+\sqrt{10r+r^2})^2=144\\
8+r+\sqrt{10r+r^2} = 12 \implies \sqrt{10r+r^2} = 4-r \implies 10r+r^2 = 16-8r+r^2\\
18r = 16 \therefore \boxed{r = \frac{8}{9}}





[/tex3]