A rhombus has all sides of length 10 cm and one of its diagonals is 12 cm long. What is the area of the rhombus in cm²?

Introduction:
This question involves properties of a rhombus, a special type of parallelogram with all sides equal. You are given the side length and one diagonal and asked to find the area. The key fact is that the diagonals of a rhombus are perpendicular bisectors of each other, which lets us use right triangles and Pythagoras theorem to find the second diagonal and then the area.

Given Data / Assumptions:

• Each side of the rhombus is 10 cm.
• One diagonal has length 12 cm.
• We must find the area of the rhombus in cm².

Concept / Approach:
For a rhombus with diagonals d₁ and d₂, the area is given by:
Area = (1/2) * d₁ * d₂.The diagonals are perpendicular and bisect each other, so each side of the rhombus forms a right triangle with half of each diagonal as the legs and the side as the hypotenuse. Using the Pythagorean theorem on one of these right triangles allows us to find the unknown diagonal d₂.

Step-by-Step Solution:
Let the given diagonal be d₁ = 12 cm.Then each half of this diagonal is d₁/2 = 6 cm.Let the other diagonal be d₂, so each half is d₂/2.In a right triangle formed by halves of the diagonals and a side, we have:(side)² = (d₁/2)² + (d₂/2)².Given side = 10 cm, so:10² = 6² + (d₂/2)².100 = 36 + (d₂/2)² ⇒ (d₂/2)² = 64.d₂/2 = 8 ⇒ d₂ = 16 cm.Now area = (1/2) * d₁ * d₂ = (1/2) * 12 * 16 = 96 cm².

Verification / Alternative check:
Another way is to think of the rhombus as composed of four congruent right triangles, each with legs 6 cm and 8 cm (since halves of the diagonals are 6 and 8). Area of each triangle is (1/2) * 6 * 8 = 24 cm², and four such triangles give 4 * 24 = 96 cm², confirming the result.

Why Other Options Are Wrong:
48 cm² and 192 cm² are exactly half or double the correct result and typically arise from missing or misplacing the 1/2 in the area formula. 144 cm² and 60 cm² are not consistent with the diagonal lengths and side length when checked through the Pythagorean relation or area of the component triangles.

Common Pitfalls:
Students sometimes confuse the formula for area of a rhombus with that of a parallelogram (base × height) and try to use side × side, which is incorrect. Others forget that the given diagonal is split in half when forming the right triangles, leading to errors in applying Pythagoras theorem.

Final Answer:
The area of the rhombus is 96 cm².