A diagonal in a quadrilateral

Question

A convex quadrilateral $ABCD$ with sides $AB=8$ cm, $BC=16$ cm, $CD=4$ cm and $AD=6$ cm is given. Find the diagonal $BD$ if the length is an integer.

Answer 1

Use the triangle inequality on $\triangle ABD$ and $\triangle BCD$: a triangle exists with sides $a,b,c$ if and only if $a+b>c$, and cyclic permutations thereof. $6+8>BD$, $16>4+BD$. Wasn't that hard, was it?

Answer 2

Let's put $$ \left\{ {\matrix{ {\mathop {AB}\limits^ \to = {\bf a}} & {\left| {\bf a} \right| = 8} \cr {\mathop {BC}\limits^ \to = {\bf b}} & {\left| {\bf b} \right| = 16} \cr {\mathop {CD}\limits^ \to = {\bf c}} & {\left| {\bf c} \right| = 4} \cr {\mathop {DA}\limits^ \to = {\bf d}} & {\left| {\bf d} \right| = 6} \cr } } \right. $$ by which $$ {\bf a + b + c + d} = {\bf 0} $$

Now we have that the diagonal equals $$ diag = \mathop {BD}\limits^ \to = - {\bf a} - {\bf d = c} + {\bf b} $$ so that for its modulus we will have $$ \eqalign{ & \left| {\mathop {BD}\limits^ \to } \right|^{\,2} = \left| {\bf a} \right|^{\,2} + \left| {\bf d} \right|^{\,2} + 2{\bf a} \cdot {\bf d} = \left| {\bf c} \right|^{\,2} + \left| {\bf b} \right|^{\,2} + 2{\bf c} \cdot {\bf b} = \cr & = 100 + 2{\bf a} \cdot {\bf d} = 272 + 2{\bf c} \cdot {\bf b} = \cr & = 100 + 2\left| {\bf a} \right|\left| {\bf d} \right|\cos \alpha = 272 + 2\left| {\bf c} \right|\left| {\bf b} \right|\cos \beta = \cr & = 100 + 96\cos \alpha = 272 + 128\cos \beta \cr} $$

Note that the angle $\alpha$ is the angle between the vectors $\mathop {AB}\limits^ \to $ and $\mathop {DA}\limits^ \to$, and therefore it is supplementary to the internal angle in $A$, and analogously for the angle $\beta$.

Since the cosine varies between $-1$ and $1$ we shall have $$ \eqalign{ & \left\{ \matrix{ 100 - 96 \le \left| {\mathop {BD}\limits^ \to } \right|^{\,2} \le 100 + 96 \hfill \cr 272 - 128 \le \left| {\mathop {BD}\limits^ \to } \right|^{\,2} \le 272 + 128 \hfill \cr} \right.\quad \Rightarrow \quad \cr & \Rightarrow \quad 144 \le \left| {\mathop {BD}\limits^ \to } \right|^{\,2} \le 196\quad \Rightarrow \quad 12 \le \left| {\mathop {BD}\limits^ \to } \right| \le 14 \cr} $$

Therefore we get the following possible results

$$ \matrix{ {\left| {\mathop {BD}\limits^ \to } \right| = 12} & {\cos \alpha = 44/96} & {\cos \beta = - 1} \cr {\left| {\mathop {BD}\limits^ \to } \right| = 13} & {\cos \alpha = 69/96} & {\cos \beta = - 103/128} \cr {\left| {\mathop {BD}\limits^ \to } \right| = 14} & {\cos \alpha = 1} & {\cos \beta = - 76/128} \cr } $$