A chord is a line connecting any two points on a circle. This geometric principle can be used to lay out a line at a 90° angle to a base line. This method is very

FIGURE 13.1. Laying out a right angle by the chord method.

simple and can be accomplished with two different lengths of string or even tree branches. The one disadvantage is that the base line must be extended past the turning point (B). This method is illustrated in Figure 13.1 The line BD is established at a 90° angle by completing the following steps:

1. Establish the base line AC if it is not in existence (fence or road edge, etc.).

2. Establish the vertex at B.

3. Set points X and Y equidistant from point B and on line AC.

4. Use a length greater than the distance XB or YB for Z, and scribe an arc from X and Y as shown.

5. Set a stake at the intersection of the two arcs (D).

6. The line established by this stake and B will be at 90° with the base line AC.

This method is simple in principle but not easy to complete because of the difficulty in marking the arcs. On tilled ground they can be formed by marking the surface, but when working on grass or taller vegetation it is much more difficult. An alternative is to use two tape measures. Attach one at X and the other at Y. Any point where both tapes have the same reading will be on a 90 degree angle from the base line.

The 3-4-5 method of laying out a right angle is based on the Pythagorean theorem: for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. In this method, any multiples of 3 and 4 used as the sides of a right triangle will result in the hypotenuse being a multiple of 5. To prove this, study the following equation:

Shown graphically, Figure 13.2.

This method has three requirements: (1) the same units (feet, yards, etc.) must be used on all three sides; (2) the same multiples of 3, 4, and 5 are used for the

lengths of the three sides; (3) the longest length is used for the hypotenuse. When these three requirements are met, a 90° angle will be established.

When using the 3-4-5 method to lay out a right angle the easiest way is to use three people and two tape measures. When two tapes are used, either the 3 or the 4 dimension is used as the base line, and the two corners on the base line are marked. Then two people, one standing at each base line corner, hold two tape ends together at the correct dimensions, and a third person holds the remaining tape ends together at the correct dimensions and moves the third corner until the tapes are all at equal tension.

This process also may be accomplished by using a 100-ft tape. Because surveyor's steel tapes are not designed to be bent at a sharp angle, loops must be formed at two of the corners. It is recommended that at least a 5-ft loop be used. Study Figure 13.3.

By completing the following steps, a 90° angle may be laid out using the 3-4-5 method. This procedure will require three people.

1. Establish the base line (AB) and corner B.

2. Lay out the tape along the base line with the 20-ft mark (4 ft x 5 ft) at corner B and the zero mark at corner A.

3. Set corner A, and have a person hold the zero mark on the corner.

4. Form a 5-ft loop in the tape, and have a person hold the 20-ft mark over the 25-ft mark, and align these marks over corner B.

5. Lay out the remaining tape in the direction of corner C.

6. Note the position of the 40-ft mark (25 ft + 15 ft) (15 ft = 3 ft x 5), and form a 5-ft loop in the tape. Hold the 40 and 45 ft marks on corner C.

7. Extend the tape back to corner A.

8. Hold the 70 ft (45 ft + 25 ft) (25 ft = 5 ft x 5) and 0 foot marks together at corner A.

9. If the individuals at A and B hold their positions carefully on the baseline while the individual at C tightens the tape in both directions, a 90° angle will be made at B.

This process will work for any combination of lengths as long as they are multiples of 3, 4, and 5. One advantage of this method is that the base line does not need to extend past the 90° corner.

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