Question

Let G be the centroid of the LABC with sides length a, b, c. If Pbe any point in the plane of AABC such that PA = 2, BP = 5, PC = 6 and PG = 4. then find a^2+v^2+c^2​

Answer 1

Answer:

Let the coordinates of A, B and C be (x1.y1), (x2.y2) and (x3.y3)

Then coordinates of centroid G will be ((x1­ + x2 + x3)/3, (y1 + y2 + y3)/3 )

Let us the assume that centroid lies on the origin

⇒ x1­ + x2 + x3 = 0 and y1 + y2 + y3 = 0

Let the coordinates of point O be (x, y)

We have to prove OA2 + OB2 + OC2 = GA2 + GB2 + GC2 + 3GO2

Now, L.H.S. = OA2 + OB2 + OC2

(x – x1)2 + (y – y1)2 + (x – x2)2 + (y – y2)2 + (x – x3)2 + (y – y3)2

= 3x2 + 3y2 + x12 + x22 + x32 + y12 + y22 + y32 – 2x (x1 + x2 + x3) – 2y(y1 + y2 + y3)

= 3x2 + 3y2 + x12 + x22 + x32 + y12 + y22 + y32

R.H.S. = GA2 + GB2 + GC2 + 3GO2

= (x1 – 0)2 + (y1 – 0)2 + (x2 – 0)2 + (y2 – 0)2 + (x3 – 0)2 + (y3 – 0)2 + 3[(x – 0)2 + (y – 0)2]

= x12 + y12 + x22 + y22 + x32 + y32 + 3(x2 + y2)

= 3x2 + 3y2 + x12 + x22 + x32 + y12 + y22 + y32

= L.H.S.

@jannat

Answer 2

Hey guy dont ask questions from pre rmo test .I know you are fron aakash.