Perpendicular by Pythagorean Theorem

The easiest and most accurate way to locate points on a line or to turn a given angle, such as 90°, from one line to another is to use a surveying instrument called a transit. If you do not have a transit, you can locate the corner points with tape measurements by applying the Pythagorean Theorem.

1. Stretch a cord from monument A to monument B.
2. Locate points C and D by tape measurements from A.
3. If you examine Figure 25, you will observe that straight lines connecting points C, D, and E form a right triangle with one side 40 feet long and the adjacent side 35 feet long. By the Pythagorean Theorem, the length of the hypotenuse of this triangle, the line ED, is equal to the square root of 352 + 402, which is approximately 53.1 feet. Because figure EG DC is a rectangle, the diagonals both ways, ED and CG, are equal. The line from C to G should also measure 53.1 feet.
4. Have one person hold the 53.1 foot mark of a tape on D, have another hold the 35 foot mark of another tape on C, and have a third person walk away with the joined 0 foot ends. When the tapes come taut, the joined 0 foot ends will lie on the correct location for point E. The same procedure, but this time with the 53.1 foot length of tape running from C and the 35 foot length running from D, will locate corner point G. Corner points F and H can be located by the same process, or by extending CE and DG 20 feet.

The equation for the Pythagorean Theorem is as follows:

C² = A² + B²

C is the hypotenuse that you are solving for. A and B are the lengths of the two known sides. When you solve for C, you get the following formula:

C=√A² + B²