Answer:

In a rhombus, the diagonals bisect each other at right angles and divide the rhombus into four congruent right triangles. Let's denote the length of one side of the rhombus as "s" and the lengths of the diagonals as "d1" and "d2."

Given that the length of one side is 20 cm, we have s = 20 cm.

Also, we know that the diagonals of the rhombus are a cm and 32 cm. Therefore, we have d1 = a cm and d2 = 32 cm.

In each right triangle formed by the diagonals, the hypotenuse is one of the diagonals, and the legs are half the lengths of the diagonals. Let's consider one of the right triangles.

The hypotenuse of the right triangle is d1 (a cm), and the legs are half the lengths of the diagonals, which are a/2 and 16 cm (32 cm divided by 2).

Applying the Pythagorean theorem, we can find the relationship between the sides of the right triangle:

(hypotenuse)^2 = (leg1)^2 + (leg2)^2

(a cm)^2 = (a/2)^2 + (16 cm)^2

Simplifying the equation:

a^2 = (a^2)/4 + 256

Multiplying both sides of the equation by 4 to eliminate the fraction:

4a^2 = a^2 + 1024

3a^2 = 1024

Dividing both sides by 3:

a^2 = 1024/3

Taking the square root of both sides:

a = √(1024/3)

a ≈ 18.85 cm (rounded to two decimal places)

Therefore, the value of "a" is approximately 18.85 cm.

Verified answer

Answer:

24cm

The steps are given in the attachment

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