In `Delta ABC,AB=9,AC=17.5`, altitude from A to line BC cut at M, AM = 3. Then

Options

A. radius of circle which circumscrive `Delta ABC` is 26.25
B. radius of circle which which circumscribe `Delta ABM` is 4.5
C. orthocentre of `Delta ABC` lies outside `Delta ABC`
D. orthocentre of `Delta ABC` lies inside `Delta ABC`

Answer

A, B, C

Explanation

From the figure, `sin theta =(1)/(3)`
In `Delta ABC`, using sine rule `(17.5)/(sin theta)=2R rArr R = 26.25`
In `Delta ABM`, using sine rule `(3)/(sin theta)=2R_(1)rArr R_(1)=4.5`, where `R_(1)` is circumradius of `Delta ABM`.
Also, `cos A=(17.5^(2)+9^(2)-BC^(2))/(2(17.5)(9))`
`=(17.5^(2)+9^(2)-(CM+MB)^(2))/(2(17.5)(9))`
`17.5^(2)+9^(2)-(17.5^(2)-9)-(9^(2)-9)`
`=(-2sqrt(17.5^(2)-9)sqrt(9^(2)-9))/(2(17.5)(9))`
`lt 0`
`therefore A` is obluse angle
`therefore` orthocenter lies outside `Delta ABC`.