The diagonal of the floor of a rectangular closet is $7\frac{1}{2}$ feet. The shorter side of the closet is $4\frac{1}{2}$ feet. What is the area of the closet in square feet?

### Concept & Pythagorean Theorem The floor of a rectangular closet forms a right-angled triangle with its length, width (shorter side), and diagonal. We can find the missing length using the Pythagorean theorem: $$ \text{Diagonal}^2 = \text{Length}^2 + \text{Width}^2 $$ Once the length is found, the area of the rectangle is: $$ \text{Area} = \text{Length} \times \text{Width} $$ ### Step-by-Step Solution 1. Convert the given mixed fractions into improper fractions to make calculations easier: Diagonal ($d$) = $7\frac{1}{2} = \frac{15}{2}$ feet Width ($w$) = $4\frac{1}{2} = \frac{9}{2}$ feet 2. Let the length of the closet be $l$. Apply the Pythagorean theorem: $$ \left(\frac{15}{2}\right)^2 = l^2 + \left(\frac{9}{2}\right)^2 $$ $$ \frac{225}{4} = l^2 + \frac{81}{4} $$ 3. Isolate $l^2$ by subtracting $\frac{81}{4}$ from both sides: $$ l^2 = \frac{225}{4} - \frac{81}{4} = \frac{144}{4} $$ 4. Simplify the fraction and find $l$: $$ l^2 = 36 $$ $$ l = 6 \text{ feet} $$ 5. Calculate the area of the rectangular closet: $$ \text{Area} = l \times w = 6 \times \left(\frac{9}{2}\right) $$ $$ \text{Area} = 3 \times 9 = 27 \text{ sq. feet} $$ ### Exam Strategy & Shortcut Recognize Pythagorean triples. The sides $l$, $\frac{9}{2}$, and $\frac{15}{2}$ can be scaled by multiplying by 2 to get $2l$, $9$, and $15$. Notice that dividing $9$ and $15$ by $3$ gives $3$ and $5$. This means the sides form a standard $3-4-5$ right triangle multiplied by a scale factor of $3$. So, the missing scaled side is $4 \times 3 = 12$. Dividing by the initial multiplier of $2$ gives $l = \frac{12}{2} = 6$. This avoids fractions entirely. ### Common Pitfall A common mistake is trying to square the mixed fractions directly (e.g., squaring $7\frac{1}{2}$ as $49\frac{1}{4}$), which violates basic algebraic rules. Always convert to improper fractions first. ### Final Answer Therefore, the correct answer is **$27$**.