Question

The three non-parallel edges of a rectangular solid are in the ratio 1 : 2 : 3. The diagonal of the solid is 2√14m. Find the length of the longest side.

Answer 1

let's suppose that the edges of the solid are a, b, c and the diagonal is d. The edges b and c are the sides of the solid base. The diagonal of the base of the solid is e.

Since we have ratio between the edges let's suppose the following:

$a:b:c = 1:2:3; a = x, b= 2x, c= 3x$

Let's find the diagonal of the base e using the Pythagorean theorem:

$e^2 = a^2 + b^2 ; e^2= 4x^2 + 9x^2$

$e^2 = 13x^2$

Let's use the Pythagorean theorem in the triangle that is build by the edge e, diagonal e and solid diagonal d:

$d^2 = c^2 + e^2$

$(2√14)^2 = x^2 + 13x^2$

$4* 14 = 14x^2$

$4 = x^2; x =2$

The question was to find the longest side:

$a= 2, b= 4, c= 6$

Answer: 6m