If in a triangle `PQR; sin P, sin Q, sin R` are in A.P; then (A)the altitudes are in AP (B)the altitudes are in HP (C)the altitudes are in GP (D)the medians are in AP

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.
.