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Difficulty: MediumTriangles and Pythagorean Theorem

The floor of a rectangular storage room has a diagonal measurement of 3939 feet and a width of 1515 feet. What is the total area of the floor in square feet?

Answer: 540 square feet

Answer

540
The storage room floor forms a right triangle with the 39-foot diagonal as the hypotenuse and the 15-foot width as a leg. The missing length is calculated to be 36 feet using the Pythagorean theorem. Multiplying this 36-foot length by the 15-foot width yields a total area of 540 square feet.

Step-by-Step Solution

1
Identify the right triangle formed by the floor dimensions
A right triangle is formed with the diagonal (39 feet) as the hypotenuse and the width (15 feet) as one leg.
The diagonal of any rectangle splits it into two identical right triangles.
2
Find the missing length of the floor
The length is 36 feet.
Using the Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2), 152+b2=39215^2 + b^2 = 39^2. This simplifies to 225+b2=1521225 + b^2 = 1521, so b2=1296b^2 = 1296 and b=36b = 36. Alternatively, recognize the 5-12-13 Pythagorean triple multiplied by 3.
3
Calculate the area of the rectangle
Area = 540 square feet.
The area of a rectangle is its length multiplied by its width (36×15=54036 \times 15 = 540).

Key Concept

Using the Pythagorean theorem to find unknown dimensions for area calculations.

Alternative Method

Instead of calculating 39239^2 and 15215^2 manually, recognize that the given sides are multiples of the common 5-12-13 Pythagorean triple (3×5=153 \times 5 = 15 and 3×13=393 \times 13 = 39). This immediately gives the missing length as 3×12=363 \times 12 = 36.
Estimated Time:1m 0s
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Topics

Triangles and Pythagorean Theorem

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