Two sides of a parallelogram have lengths 10 and 12. What can you say about the lengths of the diagonals?

A parallelogram is a simple quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure.

Consider that an isosceles triangle ABC as below:

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

$\begin{array}{l}AB+BC>AC\\ AB+AC>BC\\ AC+BC>AB\end{array}$

The parallelogram can be drawn as shown below:

As it can be seen that the two sides are given so using the above figure, therefore,

$$\begin{gathered} AB+BC>AC \\ 10+12>AC \\ 22>AC \\ AC<22 \end{gathered}$$

Similarly finding the BD, using the same theorem,

$$\begin{gathered} AB+AD>BD \\ 10+12>BD \\ 22>BD \\ BD<22 \end{gathered}$$

Thus, it can be observed that the diagonals of the parallelogram will have a value less than 22.

Therefore, $AC<22$and $BD<22$.