The diagonal of the floor of a rectangular closet is 7.5 feet and the shorter side of the closet is 4.5 feet. What is the area of the closet in square feet?

Introduction / Context:
This question involves a rectangular floor, its diagonal, and a side length. Problems like this are common in coordinate geometry and mensuration for aptitude tests. The key idea is that the diagonal of a rectangle forms the hypotenuse of a right angled triangle with the length and breadth as the other two sides. Once both side lengths are known, the area of the rectangle can be calculated easily as length multiplied by breadth.

Given Data / Assumptions:
• The floor is rectangular.• The diagonal of the floor = 7.5 feet.• The shorter side (breadth) = 4.5 feet.• We assume standard right angle rectangle properties so that Pythagoras theorem applies.

Concept / Approach:
In a rectangle, if one side is a, the other side is b, and the diagonal is d, then Pythagoras theorem gives d^2 = a^2 + b^2. Here, we know d and one side, so we can solve for the unknown side. After we find both side lengths, the area A of the rectangle is A = length * breadth. The main skills tested are the use of Pythagoras theorem and careful handling of squares and square roots.

Step-by-Step Solution:
Step 1: Let the longer side be L and the shorter side be B.B = 4.5 ft, d = 7.5 ft.Step 2: Apply Pythagoras theorem.d^2 = L^2 + B^2.7.5^2 = L^2 + 4.5^2.Step 3: Compute the squares.7.5^2 = 56.25 and 4.5^2 = 20.25.Step 4: Rearrange to find L^2.L^2 = 56.25 − 20.25 = 36.Step 5: Take the square root.L = sqrt(36) = 6 ft.Step 6: Compute the area of the rectangular floor.Area = L * B = 6 * 4.5 = 27 square feet.

Verification / Alternative check:
We can verify that 6 ft, 4.5 ft, and 7.5 ft indeed form a right triangle. Check using Pythagoras theorem: 6^2 + 4.5^2 = 36 + 20.25 = 56.25 and 7.5^2 is also 56.25. Since both sides of the equation match, the dimensions are consistent. The area 27 square feet also makes sense because both side lengths are moderate values and their product falls within a reasonable range for such a closet.

Why Other Options Are Wrong:
Option A: 5.5 is far too small and cannot be a product of two lengths that are both greater than 4 feet.Option B: 13.5 would correspond to length 3 ft and breadth 4.5 ft, which would not give the correct diagonal of 7.5 ft.Option D: 37 square feet is too large for the given side and diagonal and does not match the computed product of the side lengths.

Common Pitfalls:
Learners sometimes confuse which value is the diagonal and which ones are the sides, or they may incorrectly add rather than subtract when applying Pythagoras theorem. Another issue is rounding too early or miscalculating simple squares like 7.5^2. Writing each step clearly and verifying each arithmetic operation helps avoid these mistakes in exams.

Final Answer:
The area of the closet floor is 27 square feet.