Question

Question #a585d

Answer 1

option (1) 1:2:3

Explanation:

From Fig

• Radius of the incircle (r)
  `r/(a/2)=tan30=1/sqrt3` ,

where a = length of the side of the equilateral triangle.

`=>r=a/(2sqrt3).......color(red) "(1)"`

• Radius of the circumcircle (R)

`R/(a/2)=sec30=2/sqrt3`

`=>R=a/sqrt3...... ..color(red) "(2)"`

• Radius of the excircle `(r_1)`

`r_1/a=sin60=sqrt3/2`

`=>r_1=(sqrt3a)/2...... ..color(red) "(3)"`

So from (1) ,(2) and (3) the ratio of three radii

`r:R:r_1=a/(2sqrt3):a/sqrt3:(sqrt3a)/2=1:2:3`

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Alternative way

We know, for an equilateral triangle radius of incircle (r) is half of the radius of the circumcircle (R) ( Since two radii meet at the centroid of the equilateral triangle and sum of their length is the length of the median)
So r = r and R = 2r
Now Area of the triangle `Delta =s*r`,
where `s = 1/2*"perimeter"=(3a)/2`,(if a = length of the side of the equilateral triangle.)

Now radius of the excircle is `(r_1)` and `(r_1)(s-a)=Delta = sr`

Therefore `(r_1)= (sr)/(s-a)=(("3a"/2)r)/("3a"/2-a)=(("3a"/2)r)/(a/2)=3r`

So the ratio `r:R:r_1=r:2r:3r = "1:2:3"`

For formula please see
https://en.wikipedia.org/wiki/Incircle_and_excircles_of_a_triangle