In an equilateral triangle, the inradius, circumradius, and one of the exradii are in the ratio

Monjur Alahi

In an equilateral triangle, the inradius, circumradius, and one of the exradii are in the ratio
A. `2: 4:5`
B. `1:2:3`
C. `1:2:4`
D. `2:4:3`


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Salim Ahmed Kawsar

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Correct Answer - B
We have
`Delta = (sqrt3)/(4) a^(2), s = (3a)/(2)`
`:. r = (Delta)/(s) = (a)/(2sqrt3), R = (abc)/(4Delta) = (a^(3))/(sqrt3 a^(2)) = (a)/(sqrt3)`
and `r_(1) = (Delta)/(s-a) = (sqrt3//4a^(2))/(a//2) = (sqrt3)/(2) a`
Hence, `r:R:r_(1) = (a)/(2sqrt3) :(a)/(sqrt3): (sqrt3)/(2) a = 1:2:3`

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