Let side a,b and c of `DeltaABC` be related by the relation a : b : c = 3 : 5 : 4. Altitudes AD,BE and CF are dropped on BC, CA and AB, respectively. If `P_(1)D+P_(2)E+P_(3)F=42`, then the value of a + b + c is

The three vectors a, b and c with magnitude 3, 4 and 5 respectively and a+b+c=0, then the value of a.b+b.c+c.a is

The three vectors a, b and c with magnitude 3, 4 and 5 respectively and a+b+c=0, then the value of a.b+b.c+c.a is

The three vectors a, b and c with magnitude 3, 4 and 5 respectively and a+b+c=0, then the value of a.b+b.c+c.a is

In DeltaABC and DeltaDEF , AB= DE, AB abs DE, BC= EF and BC abs EF . Vertices A, B and C are joined to vertices D, E and F respectively. Show that AD abs CF and AD = CF

Let A and B are events of an experiment and P(A)=1/4,P(A uu B)=1/2, then value of P(B/A^(c)) is 2/3 b.1/3 c.5/6 d.1/2

Let ABC and A'B'C' be two triangles in which AB gt A'B', BC gt B'C' and CA gt C'A' . Let D, E and F be the midpoints of the sides BC, CA and AB respectively. Let D', E' and F' be the midpoints of the sides B'C', C'A' and A'B' respectively. Consider the following statements : Statement I: AD gt A'D', BE gt B'E' and CF gt C'F' are always true. Statement II : (AB^(2)+BC^(2)+CA^(2))/(AD^(2)+BE^(2)+CF^(2)) =(A'B'^(2)+B'C'^(2)+C'A'^(2))/(A'D'^(2) +B'E'^(2) +C'F'^(2)) Which one of the following is correct in respect of the above statements?

Let A, B, and C be the vertices of a triangle. Let D, E, and F be the midpoints of the sides BC, CA, and AB respectively. Show that vec(AD)+vec(BE)+vec(CF)=vec(0) .

Let s denote the semi-perimeter of DeltaABC where BC=a, CA=b , and AB=c . If a circle touches the sides BC, CA, and AB at D, E, and F respectively, prove that BD= s-b .