If in a triangle `PQR; sin P, sin Q, sin R` are in A.P; then

Options

A. the altitudes are in AP
B. the altitudes are in HP
C. the medians are in GP
D. the medians are in AP

Answer

B

Explanation

Let triangle be the area of `DeltaPQR`.
`:. Delta =(1)/(2)xxpxxh_(1)" " [h_(1),h_(2),h_(3) " are altitudes "]`
`implies h_(1)=(2Delta)/(p) " " "….(i)"`
Similarly, `h_(2)=(2Delta)/(q)" " "….(ii)"`
and `h_(3)=(2Delta)/(r )" " "....(iii)"`
According to the question, sinP, sinQ, sin R are in AP.
Then, `kp,kq,kr, " are in AP " " " [" by sine rule "]`
` implies p,q,r` are in AP.
`implies (2Delta)/(h_(1)),(2Delta)/(h_(2)),(2Delta)/(h_(3)) " are in AP. " " "[" by Eqs. (i),(ii) and (iii) "]`
`implies (1)/(h_(1)),(1)/(h_(2)),(1)/(h_(3))` are un AP.
`implies h_(1),h_(2),h_(3)` are in HP.
`implies ` Altitudes are in HP.
.