Explanation:

A rhombus is a quadrilateral with all sides equal in length, and opposite sides are parallel. The diagonals of a rhombus are line segments that connect opposite vertices of the rhombus.

Properties of the diagonals of a rhombus:

1. The diagonals of a rhombus bisect each other at 90 degrees.
2. The length of each diagonal is different.
3. The diagonals of a rhombus are perpendicular bisectors of each other.
4. The diagonals of a rhombus divide the rhombus into four congruent triangles.

Proof that the diagonals of a rhombus have different lengths:

Let ABCD be a rhombus with diagonals AC and BD. We need to prove that AC and BD have different lengths.

In rhombus ABCD, AB = BC = CD = DA (by definition of a rhombus).

Let E be the midpoint of AC, and F be the midpoint of BD.

Then, AE = EC and BF = FD (by definition of midpoint).

Also, AF = FB (by definition of a rhombus).

Now, consider triangle AEF and triangle BEF.

AE = EC, BF = FD, and AF = FB (as proved above).

Therefore, by the Side-Side-Side (SSS) congruence criterion, triangle AEF is congruent to triangle BEF.

So, EF = EF (by CPCTC - Corresponding Parts of Congruent Triangles are Congruent).

Since EF is a line segment that connects two opposite sides of the rhombus, it is a diagonal of the rhombus.

Therefore, we have proved that the diagonals of a rhombus have different lengths.

Hence, the correct answer is option C - of different length.