A courtyard in the shape of a right triangle is set in the middle of three square office buildings, with one office building along each side. Which statement is true about the three office buildings?
The area of Building C minus the area of Building B equals the area of Building A.
The area of Building C minus the area of Building A is equal to twice the area of Building B.
The area of Building B minus the area of Building C is equal to the area of Building A.
The area of Building B plus the area of Building C is equal to twice the area of Building A.
To determine the correct statement about the three office buildings surrounding the courtyard shaped like a right triangle, we need to analyze their areas in relation to the triangle’s dimensions.
Let’s denote the sides of the right triangle as a (for Building A), b (for Building B), and c (for Building C, where c is the hypotenuse). The area of a square building can be calculated as the square of its side length.
1. The area of Building A = a²
2. The area of Building B = b²
3. The area of Building C = c²
Since the triangle is a right triangle, we apply the Pythagorean theorem:
c² = a² + b²
Now, let’s evaluate the statements:
– Statement 1: The area of Building C minus the area of Building B equals the area of Building A.
( c² – b² = a² ) (From ( c² = a² + b² ), this is not true.)
– Statement 2: The area of Building C minus the area of Building A is equal to twice the area of Building B.
( c² – a² = 2b² ) (Using the Pythagorean theorem, this also does not hold.)
– Statement 3: The area of Building B minus the area of Building C is equal to the area of Building A.
( b²