Home
Class 12
MATHS
If p(1),p(2),p(3) are the altitudes of a...

If `p_(1),p_(2),p_(3)` are the altitudes of a triangle from the vertieces A,B,C and `Delta` is the area of the triangle then prove that `(1)/(p_(1))+(1)/(p_(2))-(1)/(p_(3))=(2ab)/((a+b+c)Delta)"cos"^(2)(C)/(2)`

Text Solution

Verified by Experts

Since `Delta=(1)/(2)ap_(1)=rArr (1)/(p_(1))=(a)/(Delta)`
Similarly, `(1)/(p_(2)=(b)/(2Delta),(1)/(p_(3))=(c)/(2Delta)`
`:. (1)/(p_(1))+(1)/(p_(2))+(1)/(p_(3))+(1)/(2Delta)(a+b-c)`
`=(2(s-c))/(2Delta)=(s-c)/(Delta)=(s(s-c))/(ab)*(ab)/(sDelta)`
`=(ab)/(((a+b+c)/(2))Delta)*"cos"^(2)(C)/(2)`
`=(2ab)/((a+b+c)Delta)"cos"^(2)(C)/(2)`
Promotional Banner

Similar Questions

Explore conceptually related problems

If p_(2),p_(2),p_(3) are the perpendiculars from the vertices of a triangle to the opposite sides, then prove that p_(1)p_(2)p_(3)=(a^(2)b^(2)c^(2))/(8R^(3))

If the lengths of the perpendiculars from the vertices of a triangle ABC on the opposite sides are p_(1), p_(2), p_(3) then prove that (1)/(p_(1)) + (1)/(p_(2)) + (1)/(p_(3)) = (1)/(r) = (1)/(r_(1)) + (1)/(r_(2)) + (1)/(r_(3)) .

Find the area of the triangle with vertices A(1,1,2)B(2,3,5) and C(1,5,5).

Find the area of the triangle whose vertices are A (3,-1,2), B(1,-1,-3) and C(4,-3,1).

If the angles A,B,C of a triangle are in A.P. and sides a,b,c, are in G.P., then prove that a^2, b^2,c^2 are in A.P.

IF in a Delta ABC , the altitudes from the vertices A,B,C on opposite sides are in H.P , then sin A , sin B , sin C are in

The area of the triangle whose vertices are A(1,-1,2),B(2,1-1)C(3,-1,2) is …….

If the area of the triangle formed by the vertices (p,p),(5,6),(5,-2) is 32 sq. units. Find the value of p.

Find the area of the triangle whose vertices are A(3, -1,2) ,B(1, - 1, 3) and C(4, - 3 , 1)