To determine whether two interior walls meet at a right angle

Question

To determine whether two interior walls meet at a right angle, carpenters often mark a point 3 feet from the corner on one wall and a point 4 feet from the corner (at the same height) on the other wall. If the straight-line distance between those points is exactly 5 feet, the walls are square. At what angle do the walls meet if the distance measures 5 feet 3 inches? Round to the nearest tenth of a degree.

Answer 1

The question introduces the concept of the 3-4-5 right triangle to determine if the wall is at at a right angle. Marking a point on one wall at 3 feet, and on another wall at 4 feet is forming the legs of the triangle with the corner of the wall as the common corner shared by both legs. Measuring the point distance between the two points will measure the hypotenuse of the right triangle. If the lengths are exactly 3-4-5, this will form the common 3-4-5 right triangle.

However, if the walls meet at 5 feet and 3 inches then it is no longer a right triangle. The problem has given the length of every leg of the triangle: 3 feet, 4 feet, and 5 feet 3 inches (5.25 feet). The law of cosines can be applied directly to solve for the angle. Using a = 3 feet, b = 4 feet, and c = 5.25 feet, we desire the angle common to the walls a and b. This is the corner opposite of the straight-line distance c, so we denote this by angle C. c^2 = a^2 + b^2 - 2ab cosC, then C = 96.1 degrees.