Home
Class 12
MATHS
If in a triangle PQR; sin P, sin Q, sin ...

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 medians are in GP

D

the medians are in AP

Text Solution

Verified by Experts

The correct Answer is:
B

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.
.
Doubtnut Promotions Banner Mobile Dark
|

Similar Questions

Explore conceptually related problems

If any triangle A B C sin ^(2) A+sin ^(2) B-sin ^(2) C=

In a triangle PQR with right angle at Q , the value of angle P is x, PQ = 7cm and QR = 24 cm , then find sin x and cos x

Knowledge Check

  • In any triangle ABC , sin.(A)/(2) is

    A
    less than `(b+c)/(2)`
    B
    greater than `(2a)/(a+b+c)`
    C
    less than or equal to `(a)/(b+c)`
    D
    None of these
  • With the usual notation in the triangle A B C [[ 1 , 1 , 1 ],[ sin A , sin B , sin C ],[ sin ^2 A , sin ^2 B , sin ^3 C ]] =

    A
    `(1)/(8 R^3)(a-b)(a-c)(b-c)`
    B
    `8 R^3`
    C
    `(a-b)(b-c)(c-a)`
    D
    `(1)/(8 R)(a-b)(a-c)(b-c)`
  • If a, b, c are in A.P., p, q, r are in HP and ap, bq, cr are in GP, then (P)/(r)+(r)/(P) is equal to

    A
    `(a)/(c)-(c)/(a)`
    B
    `(a)/(c)+(c)/(a)`
    C
    `(b)/(q)+ (q)/(b)`
    D
    `(b)/(q)-(q)/(b).`
  • Similar Questions

    Explore conceptually related problems

    Prove that in any triangle ABC, (sin A)/a= (sin B)/b=(Sin C)/c .

    If the sides of a right angled triangle are in A.P then the sines of the acute angles are

    If A B C are the angles of a triangle then sin ^(2) A+sin ^(2) B+sin ^(2) C-2 cos A cos B cos C

    In a triangle PQR, /_ R = pi/2 If tan(p/2) and tan(Q/2) are the roots of ax^(2) + bx + c = 0 , a + != , then :

    In a DeltaPQR 3sin p + 4cosQ = 6 and 4sinQ + 3cosP = 1 , then the angle R is equal to :