The diagonals of a rhombus measure 24 cm and 10 cm. What is the perimeter of this rhombus (in cm)?

Introduction / Context:
This question is about a rhombus, a special type of parallelogram in which all sides are equal. A key geometric property of a rhombus is that its diagonals intersect at right angles and bisect each other. This property allows us to use right triangle relationships to find the length of a side when the diagonals are known. Once the side length is determined, calculating the perimeter simply involves multiplying the side length by four. Such problems test both your knowledge of properties of quadrilaterals and your ability to apply the Pythagoras theorem in a straightforward manner.

Given Data / Assumptions:
• The figure is a rhombus with all four sides equal in length.
• Length of one diagonal = 24 cm.
• Length of the other diagonal = 10 cm.
• Diagonals of a rhombus intersect at right angles and bisect each other.
• We need to find the perimeter of the rhombus in centimetres.

Concept / Approach:
When the diagonals of a rhombus intersect, they cut each other into two equal halves and meet at right angles. Hence, each half of the first diagonal is 24 / 2 = 12 cm, and each half of the second diagonal is 10 / 2 = 5 cm. Each side of the rhombus is the hypotenuse of a right triangle formed by these half diagonals. By applying the Pythagoras theorem in that right triangle, we can compute the side length. After that, the perimeter is simply 4 times the side length, since all sides of a rhombus are equal.

Step-by-Step Solution:
Step 1: Compute half of each diagonal, since diagonals bisect each other in a rhombus: half of 24 cm is 12 cm, half of 10 cm is 5 cm. Step 2: Consider the right triangle formed by these half diagonals and one side of the rhombus. The legs of this triangle are 12 cm and 5 cm. Step 3: Apply the Pythagoras theorem to find the side length s: s^2 = 12^2 + 5^2. Step 4: Compute the squares: 12^2 = 144 and 5^2 = 25, so s^2 = 144 + 25 = 169. Step 5: Take the square root: s = √169 = 13 cm. Step 6: The perimeter of the rhombus is 4 * s = 4 * 13 = 52 cm.

Verification / Alternative check:
The 5, 12, 13 triple is a well known Pythagorean triple, which gives further confidence that the calculations are correct. With side length 13 cm, all four sides are equal and the diagonals are consistent with the right triangle constructed. The perimeter of 52 cm is therefore logically consistent with all given dimensions and properties of a rhombus. No contradictions appear when the lengths and angle conditions are checked.

Why Other Options Are Wrong:
A perimeter of 56 cm would correspond to a side length of 14 cm, for which the associated right triangle would have hypotenuse 14 and legs 12 and 5, but 12^2 + 5^2 = 169, which is 13^2, not 14^2. Similarly, perimeters 68 cm and 72 cm imply even larger side lengths, which are not compatible with the given diagonal lengths when using the Pythagoras theorem. Hence, these options do not satisfy the geometric constraints of a rhombus with diagonals 24 cm and 10 cm.

Common Pitfalls:
Some students mistakenly take the full diagonals as the legs of the triangle instead of using half diagonals, leading to a wrong side length. Others forget that the diagonals of a rhombus are perpendicular, which is essential for applying the Pythagoras theorem correctly. It is important to remember the key properties: diagonals bisect each other at right angles and sides are all equal. Drawing a quick sketch and labelling half diagonals often helps avoid these conceptual mistakes.

Final Answer:
The perimeter of the rhombus is 52 cm.