Hamilton 2015 Q4
4. The points \(A\), \(B\) and \(C\) are the centres of three faces of a cuboid that meet at a vertex. The lengths of the sides of the triangle \(ABC\) are 4, 5 and 6.
What is the volume of the cuboid?
Hamilton Olympiad 2015, Question 4. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: \(90\sqrt{6}\)
Let the cuboid have centre \(O\) and dimensions \(2x \times 2y \times 2z\), as shown. Joining \(O\) to each of \(A\), \(B\) and \(C\) forms three right-angled triangles with sides \(x\), \(y\) and 4; \(y\), \(z\) and 5; and \(z\), \(x\) and 6.
Hence, using Pythagoras' Theorem, we get
Adding all three equations, we obtain
so that
Subtracting each of the first three equations in turn, we therefore get
Now \(2x\) is equal to \(\sqrt{4x^2}\) (because \(x\) is positive), with similar results for \(y\) and \(z\). So the volume of the cuboid \(2x \times 2y \times 2z\) is therefore
Solution © UK Mathematics Trust, from the Hamilton Olympiad 2015 solutions.