In rectangular prism \(ABCDEFGH\), the sum of the lengths of all of the edges is \(28\text{ cm}\), and the total surface area is \(13 \text{ cm}^2\).
What is the length of \(EC\), the diagonal of the prism?
Let \(EH=x\), \(HG=y\), and \(CG=z\). We construct \(EC\) and let \(EC=d\).
By the Pythagorean Theorem in \(\triangle
EHG\), \(EG^2 = EH^2 +
HG^2\).
By the Pythagorean Theorem in \(\triangle
EGC\), \(EC^2 = EG^2 +
CG^2\).
Therefore, \(EC^2 = EG^2 + CG^2 = EH^2 + HG^2
+ CG^2\).
That is, \(d^2=x^2+y^2+z^2\).
Since the sum of the lengths of all the edges is \(28\), then \(4x+4y+4z=28\) or \(x+y+z=7\).
Since the surface area of the prism is \(13\), we know \(2xy + 2yz + 2xz=13\).
Since we have squared terms and pair factor terms it might be helpful to expand \((x+y+z)^2\). \[\begin{aligned} (x+y+z)^2 &= (x+(y+z))^2\\ &= x^2 + 2x(y+z) + (y+z)^2\\ & = x^2 +2xy + 2xz + y^2 + 2xy + z^2\\ &= (x^2 + y^2 + z^2)+ (2xy + 2xz + 2yz) \end{aligned}\] Since \(x+y+z=7\), \(d^2=x^2+y^2+z^2\), and \(2xy + 2yz + 2xz=13\), we have \[\begin{aligned} 7^2 &= d^2 +13\\ d^2&= 49 - 13 = 36 \end{aligned}\] Since \(d>0\), we have \(d=6\). Therefore, the length of \(EC\) is \(6\text{ cm}\).