CHAPTER V

USES OF LINE SEGMENTS AND ANGLES IN FINDING UNKNOWN DISTANCES

THE CONGRUENT-TRIANGLE METHOD OF FINDING

DISTANCES

74. How to measure line segments indirectly. We have been studying line segments, angles, and some facts about triangles. We are now prepared to make a more extensive study of the triangle.

The triangle is used in ornamental work (Figs. 139, 140); in designing (Fig. 141); in construction (Fig. 142); in surveying (Fig. 143); in navigation (Fig. 144).

FIG. 142. USE OF A TRIANGLE IN THE CONSTRUCTION OF A BRIDGE

FIN

STREAM, AD

North

FIG. 144. TRIANGLE FOR DETERMINING THE EAST AND NORTH DISTANCES MADE BY A SHIP

In this chapter we shall learn to use the triangle to determine distances which cannot conveniently be measured directly, e.g., the height of a tree (Fig. 145), the distance across a river (Fig. 146); or those which cannot be measured directly at all, as the distance through a building or a hill (Figs. 147, 148).

To be sure, the height of a tree can be determined by climbing the tree and then applying a tape line. The distance across a river can be found by swimming across and measuring a line stretched from one bank to the other. A knowledge of mathematics, however,

makes this unnecessary. It saves work and time. Furthermore, by mathematics we can solve the problem even when direct measurement is impossible, as in Figures 147 and 148.

For example, in Fig. 148, to find the distance AB passing through the obstructing hill H, a point C is selected so that lines BE and AD may be laid off conveniently. A triangle ECD of the same size and shape as triangle ACB is laid off by making CE=CB, and CD CA. The length of AB is then the same as that of ED, and can therefore be found by measuring ED.

Another method of determining AB makes use of a triangle of the same shape as ABC, but not of the same size. A third method uses a right triangle. All of these methods obtain the required length without measuring the unknown line directly. Determining distances without measuring directly is called indirect

measurement.

75. Triangles of the same size and shape. In the problem of finding unknown distances it is necessary to know how to make a triangle which is exactly of the same size and shape as another triangle. One triangle is then an exact reproduction of the other. The two triangles are really the same triangle in two different positions, and one can be made to fit exactly on the other. Such triangles are called congruent triangles. This word comes from the Latin word congruere, meaning to agree. Congruent triangles have the sides and angles of one, equal to the corresponding sides and angles of the other. The symbol for congruence is, the symbol = meaning equal in size, and meaning similar in shape.

Examples of congruence are the "blue prints" of the draftsman, the reprints of an original, the “negative” plate in photography.

EXERCISES

1. The story is told that a soldier of Napoleon was commanded by him to determine the width of a river which the army had to cross. In a brief time he brought in the desired information. When asked how he had obtained his result he said that he did it by means of mathematics as follows: Standing at the point S (Fig. 149) and

S

---p'

FIG. 149

looking across the river, he lowered his head until a point P on the opposite bank was exactly in line with the rim of his hat and his eye. Keeping his head in rigid position, he turned, sighted along the shore line, and had a stake placed at P', a point on the shore, exactly in line with his eye and the rim of his hat.