Given:
- The diagonals of a rhombus measure 16 cm and 30 cm respectively.

To find:
- The perimeter of the rhombus.

Solution:

A rhombus is a quadrilateral with all sides of equal length. In a rhombus, the diagonals are perpendicular bisectors of each other, dividing the rhombus into four congruent right-angled triangles.

Step 1: Find the length of each side of the rhombus

Since the diagonals of a rhombus are perpendicular bisectors of each other, they divide the rhombus into four congruent right-angled triangles. Let's label the diagonals as AC and BD, and the intersection point as O.

AC = 16 cm
BD = 30 cm

Since the diagonals bisect each other at right angles, triangle AOB and triangle COD are right-angled triangles.

We can use the Pythagorean theorem to find the length of each side of the rhombus.

In triangle AOB:
AB² = AO² + OB²

In triangle COD:
CD² = CO² + OD²

Since the diagonals bisect each other, AO = CO and BO = DO.

Therefore, the equations become:
AB² = AO² + BO²
CD² = CO² + DO²

Substituting the values:
AB² = AO² + AO²
CD² = CO² + CO²

AB² = 2AO²
CD² = 2CO²

Now, substituting the given values:
AB² = 2(16)²
CD² = 2(30)²

AB² = 2(256)
CD² = 2(900)

AB² = 512
CD² = 1800

Taking square roots:
AB = √512
CD = √1800

AB ≈ 22.63 cm
CD ≈ 42.43 cm

Since all sides of a rhombus are equal, each side of the rhombus measures approximately 22.63 cm.

Step 2: Calculate the perimeter of the rhombus

The perimeter of any polygon is the sum of all its side lengths. Since a rhombus has four equal sides, the perimeter can be calculated by multiplying the length of one side by 4.

Perimeter = 4 * Side length

Perimeter = 4 * 22.63 cm

Perimeter ≈ 90.52 cm

Therefore, the perimeter of the rhombus is approximately 90.52 cm.