Given:
Perimeter of the rhombus = 100 cm
Length of one diagonal = 40 cm

To find:
Length of the other diagonal

Solution:
A rhombus is a quadrilateral with all sides equal in length. It has two pairs of parallel sides and opposite angles are equal. The diagonals of a rhombus are perpendicular bisectors of each other, meaning they intersect at right angles and divide each other into two equal parts.

Step 1:
Since the perimeter of the rhombus is 100 cm, and all sides are equal, we can divide the perimeter by 4 to find the length of each side.
Perimeter = 4 * side length
100 cm = 4 * side length
Side length = 100 cm / 4
Side length = 25 cm

Step 2:
The diagonals of a rhombus bisect each other at a right angle, dividing the rhombus into four congruent right-angled triangles.

Step 3:
Let's assume the length of the other diagonal is 'd'.
Using the Pythagorean theorem, we can find the length of the side of the rhombus in terms of 'd' and '25 cm'.
The formula for the length of the side of the rhombus is:
Length of the side = sqrt((d/2)^2 + (25 cm)^2)

Step 4:
Since the diagonals of a rhombus are perpendicular bisectors of each other, they divide the rhombus into four congruent right-angled triangles. Therefore, the sum of the squares of the lengths of the diagonals is equal to the sum of the squares of the lengths of the sides.

Using this property, we can write the equation:
(d^2 + 40 cm^2) + (d^2 + 40 cm^2) = 4*(25 cm)^2

Simplifying this equation:
2d^2 + 80 cm^2 = 4*625 cm^2
2d^2 = 2500 cm^2 - 80 cm^2
2d^2 = 2420 cm^2
d^2 = 2420 cm^2 / 2
d^2 = 1210 cm^2
d = sqrt(1210 cm^2)
d ≈ 34.8 cm

Step 5:
Therefore, the length of the other diagonal is approximately 34.8 cm.

5 cm