In a triangle $A B C, a : b : c = 4 : 5 : 6$ . The ratio of the radius of the circumcircle to that to the incircle is

15 / 4

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11 / 5

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16 / 7

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16 / 3

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Solution

The correct option is C $16 / 7$
Let $\frac{a}{4} = \frac{b}{5} = \frac{c}{6} = k R = \frac{a b c}{4 Δ} and r = \frac{Δ}{s} \Rightarrow \frac{R}{r} = \frac{a b c}{4 Δ} \times \frac{s}{Δ} = \frac{s a b c}{4 Δ^{2}} = \frac{s a b c}{4 s (s - a) (s - b) (s - c)} = \frac{4 k \times 5 k \times 6 k}{4 (\frac{15}{2} k - 4 k) (\frac{15}{2} k - 5 k) (\frac{15}{2} k - 6 k) = \frac{5 \times 6 \times 8}{7 \times 5 \times 3} = \frac{16}{7} .}$

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In a triangle ABC, a : b : c = 4 : 5 : 6. The ratio of the radius of the circumcircle to that of the incircle is

△ A B C, a : b : c = 4 : 6 : 8

a : b : c = 4 : 5 : 6

A B C, a : b : c = 4 : 5 : 6

△ A B C

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