Cuboid & Arithmetic Progression

Geometry Level 3

Cuboid $ABCDEFGH$ has a main diagonal $d = 9$ . Sides $AB$ , $BC$ and $BF$ respectively are consecutive members of an arithmetic progression $(a_n)_{n = 1}^\infty$ with a difference $k = -3$ .

Find $2|AB| - |BC| - |BF|$ .

9

11 - \sqrt{3}

3

\frac{3 + \sqrt{3}}{8}

3\sqrt{21}

\frac{1}{2} + \frac{\sqrt{2}}{2}

16

1 solution

Nick Kent
Mar 30, 2020

Let $|AB| = n$ . Then $|BC| = n - 3$ and $|BF| = n - 6$ . So, $2|AB| - |BC| - |BF| = 2n - (n - 3) - (n - 6) = \boxed{9}$

That is actually very clever. I was expecting you to do it by using $|AB|^2 + |BC|^2 + |BF|^2 = d^2$ but the way I asked for the solution allowed this shortcut. 😁

Tomáš Hauser - 1 year, 2 months ago

Cuboid & Arithmetic Progression

1 solution

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