The difference of the areas of two squares drawn on two line segments of different lengths is 32 sq. cm. Find the length of the greater line segment if one is longer than the other by 2 cm.

### Concept & Algebraic Formulation The area of a square is the square of its side. If we have two squares with side lengths $a$ and $b$, the difference in their areas is $a^2 - b^2$. We can solve for the unknown side by setting up a quadratic equation using the given constraints. ### Step-by-Step Solution 1. Let the length of the smaller line segment (side of the smaller square) be $x$ cm. 2. The length of the greater line segment is given as $(x + 2)$ cm. 3. The area of the smaller square is $x^2$. 4. The area of the larger square is $(x + 2)^2$. 5. According to the given condition, the difference in areas is 32 sq. cm: $(x + 2)^2 - x^2 = 32$ 6. Expand the squared term: $(x^2 + 4x + 4) - x^2 = 32$ 7. Simplify the equation: $4x + 4 = 32$ $4x = 28$ $x = 7$ cm 8. The length of the greater line segment is $x + 2 = 7 + 2 = 9$ cm. ### Exam Strategy & Shortcut Use option elimination directly on the greater segment. If greater is 9: smaller is 7. Areas are 81 and 49. Difference = $81 - 49 = 32$. This matches perfectly and takes seconds to compute mentally. ### Common Pitfall Solving the equation correctly to find $x = 7$, but forgetting that the question asks for the **greater** line segment, thus incorrectly choosing 7 cm as the answer. ### Final Answer Therefore, the correct answer is **9 cm**.