Final answer:
The angles formed where the front of the bookcase touches the walls are 45 degrees each. To calculate the wall space each side of the bookcase uses, we can apply the Pythagorean theorem, resulting in each side occupying approximately 38.18 inches of wall space.
Step-by-step explanation:
The question requires the application of geometry to solve for angles and wall space in a real-world scenario. Given that a bookcase fits diagonally in a corner of a room, we can visualize it as forming a right-angled triangle with the walls. Since the front of the bookcase is 54 inches wide, and it forms the hypotenuse of the triangle, we can infer that the two sides of the triangle along the walls are equal, as each side uses the same amount of wall space.
Using the properties of 45-45-90 triangles, the angles where the front of the bookcase touches the walls are 45 degrees each. Since the bookcase is set at a 45-degree angle to the walls, both of the angles between the front of the bookcase and the walls must be 45 degrees.
To calculate the amount of wall space each side uses, because we are dealing with a right-angled triangle where the two legs are equal, we can apply the Pythagorean theorem. Let x be the length of one side (along the wall). Then, we have:
- x2 + x2 = 542
- 2x2 = 2916
- x2 = 1458
- x ≈ 38.18 inches
Therefore, each side of the triangle occupies approximately 38.18 inches of wall space.