The lengths of the two diagonals of a rhombus are 24 cm and 32 cm. What is the length (in cm) of each side of the rhombus?

Introduction / Context:
This question uses a basic property of a rhombus. In a rhombus, the diagonals bisect each other at right angles, and each side of the rhombus is the hypotenuse of a right triangle formed by half of each diagonal. The problem is a direct application of Pythagoras theorem with the half diagonals as legs.

Given Data / Assumptions:

• A rhombus has diagonals of lengths 24 cm and 32 cm.
• The diagonals intersect at right angles and bisect each other.
• We need the length of a side of the rhombus.
• All sides of a rhombus are equal.
• Standard rhombus properties are assumed.

Concept / Approach:
In a rhombus, diagonals are perpendicular bisectors of each other. Therefore, each vertex of the rhombus forms a right triangle using half of each diagonal as legs and the side of the rhombus as the hypotenuse. If the diagonals are d₁ and d₂, then half diagonals are d₁ / 2 and d₂ / 2. The side length s satisfies s^2 = (d₁ / 2)^2 + (d₂ / 2)^2. We substitute the given values and compute s.

Step-by-Step Solution:
Step 1: Given diagonals are 24 cm and 32 cm, so half diagonals are 24 / 2 = 12 cm and 32 / 2 = 16 cm. Step 2: Consider the right triangle formed by these half diagonals as legs and the side of the rhombus as hypotenuse. Step 3: Let s be the side length. Then s^2 = 12^2 + 16^2. Step 4: Compute 12^2 = 144 and 16^2 = 256. Step 5: Add them to get s^2 = 144 + 256 = 400. Step 6: Therefore s = √400 = 20 cm.

Verification / Alternative check:
We can confirm by visualising the rhombus as four congruent right triangles. Each right triangle has legs 12 cm and 16 cm and hypotenuse 20 cm. This is a well known 3–4–5 type scaled triangle because 12, 16, 20 are 3, 4, 5 multiplied by 4. Since all four sides of a rhombus are equal, each side must be 20 cm, which matches the computed value.

Why Other Options Are Wrong:
Side lengths 40 cm, 60 cm or 80 cm would produce significantly larger hypotenuse values and do not match the right triangle formed by half diagonals. The value 32 cm equals one of the diagonals and would give a hypotenuse that is too long for legs 12 cm and 16 cm. Only 20 cm satisfies Pythagoras theorem with the given diagonal halves.

Common Pitfalls:
Sometimes learners mistakenly use the full diagonal lengths instead of half lengths when applying Pythagoras theorem. Another mistake is thinking diagonals of a rhombus are equal, which is only true for a square. Always remember that diagonals in a rhombus are perpendicular and bisect each other, so it is the half lengths that form the legs of the right triangle at each vertex.

Final Answer:
The side length of the rhombus is 20 cm.