the surface area and diagonal of a cuboid are 288 square cm. and 12 cm. respectively. show that the cuboid is a cube.
Answers
Let us consider the quadratic form
Q(a,b,c) = (a-b)^2+(b-c)^2+(a-c)^2 (1)
We have Q(a,b,c) = (a^2-2ab+b^2) +(b^2-2bc+c^2)+(a^2-2ac+c^2) =
= 2( a^2+b^2+c^2) - 2(ab+bc+ac) (2)
As everybody knows (or MUST know),
a^2+b^2+c^ is the square of the length of the 3D diagonal, so it is equal to cm^2.
Therefore the term 2(a^2+b^2+c^2) in (2) is equal to = 288 cm^2.
Again, as everybody knows (or MUST know),
2(ab+bc+ac) is the surface area of the rectangular prism.
Therefore, the term 2(ab+bc+ac) in (2) has the given value of 288 cm^2 under the given condition.
Now it is clear that at given data the form Q(a,b,c) is equal to zero.
But Q(a,b,c) is the sum of squares, and can be equal to zero if and only if a = b, b = c and a = c.
It implies that a = b = c and the prism is a cube under given condition.