Three Dimensional Geometry

Let the position vectors of the vertices A, B and C of a triangle be 2 i+2 j+ k, i+2 j+2 k and 2 i+ j+2 k respectively. Let l1, l2 and l3 be the lengths of perpendiculars drawn from the orthocentre of the triangle on the sides AB, BC and CA respectively, then l12+l22+l32 equals:

1. 14
2. 15
3. 13
4. 12

Answer

(4) 12

Solution

AB=BC=CA= 1+0+1= 2 equilateral triangle

∵ in an equilateral triangle orthocentre = centroid = incentre G ( 53, 53, 53 )

[figure]

l1=l2=l3= inradius = a2 3= 22 3= 1 6

∴ l12+l22+l32=3 16= 12

Answer: (4)