The diagonal of the floor of a rectangular closet is 7 1/2 feet, and the shorter side is 4 1/2 feet. What is the area of the closet floor (in square feet)?

Introduction / Context:This question tests rectangle geometry using the Pythagoras theorem. The diagonal of a rectangle forms a right triangle with the two sides as perpendicular legs. If the diagonal and one side are known, the other side can be found using: other_side = sqrt(diagonal^2 - known_side^2). Once both sides are known, area is simply side1 * side2. The values are mixed fractions (7 1/2 and 4 1/2), so converting them to decimals or improper fractions first makes calculations easier and avoids mistakes.

Given Data / Assumptions:

• Diagonal d = 7 1/2 ft = 7.5 ft
• Shorter side s = 4 1/2 ft = 4.5 ft
• Other side = sqrt(d^2 - s^2)
• Area = side1 * side2

Concept / Approach:Use the right triangle relation: d^2 = a^2 + b^2. With d and one side known, solve for the other side, then multiply both sides for area.

Step-by-Step Solution: d = 7.5, s = 4.5 Other side = sqrt(7.5^2 - 4.5^2) 7.5^2 = 56.25 4.5^2 = 20.25 Difference = 56.25 - 20.25 = 36 Other side = sqrt(36) = 6 ft Area = 4.5 * 6 = 27 square feet

Verification / Alternative check:Check the diagonal using found sides: sqrt(4.5^2 + 6^2) = sqrt(20.25 + 36) = sqrt(56.25) = 7.5, which matches the given diagonal, so the sides are correct.

Why Other Options Are Wrong: 9 and 18 are too small and would imply much smaller side lengths. 36 is too large and would require a larger second side than 6 ft. 30 comes from incorrect multiplication or wrong square subtraction.

Common Pitfalls:Squaring mixed numbers incorrectly, adding squares instead of subtracting to find the missing side, or using diagonal as if it were a side.

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