Prove that 3 times the sum of the squares of the sides = 4 times the sum of the squares of the medians of a ∆.

Question

I used the symbol to represent a triangles and not a prism. I had only one character left of the 100 allowed in the question box and I did not want to go into 'details'. So I finished it off with a triangle symbol.

Answer 1

Let the triangle be ΔABC where A, B and C are the vertices. Let the mid points of the sides BC, CA and AB be D, E and F respectively. Thus AD, CF and BE are medians. Let the point of intersection of the medians be G.

Then AD^2 = AC^2 +CD^2 –2AC.CD.cosC
AD^2 = b^2 + (a^2)/4 –ab.cosC, where b = AC, a = BC and c = AB

AD^2 = b^2 + (a^2)/4 – ab. [(b^2 +a^2 –c^2)/2ab]

AD^2 = (2b^2 +2c^2 –a^2)/4

I.e. AD = (1/2) [√ (2b^2 +c^2 –a^2)]

Similarly BE = (1/2) [√(2c^2 +2a^2 –b^2)]

And CF = (1/2) [√ (2a^2 + 2b^2 –c^2)]

Then 4(AD^2 + BE^2 +CF^2) = 4(Sum of the squares of the medians of the circle)
= 4[(1/4) [2b^2 +2c^2 –a^2] + (1/4) [2c^2 +2a^2 –b^2] + (1/4) [2a^2 + 2b^2 – c^2]]
= 2b^2 +2c^2 –a^2 +2c^2 +2a^2 –b^2 + 2a^2 + 2b^2 – c^2
= 3b^2 + 3a^2 +3c^2
= 3(a^2 + b^2 +c^2)
= 3 (Sum of the squares of the sides of the triangle)

Answer 2

Actually, a prism is 'any' figure in which the size of both of its bases is same. For eg., a right circular cylinder, a cube or cuboid- all are types of prisms. We find many uses of cylinders(jars, water tanks), cubes and cuboids(buildings, rubik's cube, boxes). The surface area of prisms is always the perimeter of its base x height.

A pyramid is a structure with its 1 base tapering to a point on the other side. For eg., right circular cone and a triangular pyramids are pyramids.

Hope that helps.

Answer 3

a= side
b= median

It's easy, only with the use of the Pythagorean Theorem:
(a/2)^2+b^2=a^2
b^2=3a^2\4
4b^2=3a^2