If in a `Delta ABC, AD, BE and CF` are the altitudes and R is the circumradius, then find the radius of the DEF.

Options

A. `R/2`
B. 2R
C. R
D. none of these

Explanation

Since BE and CF are altitudes of `Delta` ABC. Therefore, the circle described on BC as a diameter passes through E and F and its radius is `a/2`
Thus, the radius of the circumcircle of `Delta FBE` is a/2
Also, `angleFBC=90^(@)-A` `[becauseABE` is right angled at E`]`
`thereforeFE=2(a/2)sin(90^(@)-A)[becausea/(sinA)=b/(sinB)=c/(sinC)=2R]`
`rArrFE=acosA`

Let `R_(1)` be the radius of the circumcircle of `Delta DEF`. Then,
`R_(1)=(FE)/(2sinangleFDE)`
`rArrR_(1)=(acosA)/(2sin(180^(@)-2A))` `[because` Angles of pedal triangleDEF are `180^(@)-2A,180^(@)-2B` and `180^(@)-2C]`
`rArrR_(1)=(acosA)/(2sin2A)=a/(4sinA)=R/2`