CHAPTER I TRIANGLES AND PARALLELOGRAMS SECTION I GENERAL PROPERTIES OF TRIANGLES 32. After laying so much of a foundation we are ready to proceed with the task of building up the main body of geometrical knowledge, and it must be constantly borne in mind that this subject is a continuous development. We shall meet new truths at every turn, and every new truth will aid us in the establishment of still other new truths. We shall first present a few easy problems connected with the construction of triangles. DEFINITIONS 33. A triangle whose three sides are unequal in length is called a scalene triangle. A triangle which has two equal sides is called an isosceles triangle. A triangle whose three sides are all equal is called an equilateral triangle. Is an equilateral triangle also isosceles? PROPOSITION I 34. It is required to construct an equilateral triangle of which a given line-segment is one side. First, recall what an equilateral triangle is. If one side is given in length, how long are the other two sides? Now state the proposition in specific terms, thus: Let AB (see the diagram) be the given line-segment. It is required to construct an equilateral triangle, of which AB is one side. Next proceed with the construction. With centre A, and radius AB, describe a circle. These circles will intersect in two points. Why? (See Art. 25.) Let C be one of their points of intersection. Join AC and BC. NOTE. When we say 'Join AC,' we mean draw a straight line from the point A to the point C. The triangle ACB so constructed has the line-segment AB for one side; but it remains to show that this triangle is equilateral. Proof. Because A is the centre of the first circle, and the line-segments AB and AC are radii, therefore AC equals AB. Because B is the centre of the other circle, and the linesegments BA and BC are radii, therefore BC equals BA. That is, AC and BC are each equal to AB. Hence AC equals BC. (Axiom 1) Therefore AB, AC, BC are all equal, and the triangle ABC is equilateral. EXERCISES 1. Is this proposition a problem or a theorem? 2. If F is the other point of intersection of the two circles, and we join AF and BF, prove that AFB is also an equilateral triangle. 3. Join CF. Show that the triangles ACF and BCF are each isosceles. 35. The student should carefully note the order of arrangement in the preceding proposition. The same order, written out in less detail, perhaps, will be followed in all the propositions. We have 1. The general enunciation of the proposition; that is, the statement of the proposition in general terms, and the student, after reading the enunciation, should proceed no further until he thoroughly understands just what he is expected to do. 2. A re-statement of the proposition applied to a particular figure, or what is sometimes called the particular enunciation. 3. Making the necessary construction. 4. Proof that the thing constructed is what was required. In the case of a theorem the proof goes to show that the statement made in the enunciation is true. 36. In naming a circle we usually mention three points on it, these being sufficient to distinguish it from every other circle. Thus, in the diagram of this proposition, the circle whose centre is A would be called the circle BCD, and the circle whose centre is B would be called the circle ACE. EXERCISE. Would naming its centre and one point on it be sufficient to distinguish a circle from all other circles? PROPOSITION II [General Enunciation] 37. To construct a triangle two of whose sides shall be each equal to one given line-segment, and the third side equal to another given line-segment. What sort of a triangle will this be? D n E m [Particular Enunciation] Let m and n be the given line-segments. It is required to construct a triangle, of which two sides are each equal to m and the third side equal to n. [Construction] Draw anywhere in the plane a line-segment AB equal to n. With centre A, and radius equal to m, describe a circle CDF. With centre B, and radius also equal to m, describe a circle CEF. Suppose these two circles intersect at C. Do these circles necessarily intersect? If so, at how many points? Join CA and CB. Then CAB is the triangle required. [Proof] Let the pupil make a diagram for himself and letter it as in the construction. In the triangle CAB, the side AC is a radius of the circle CDF, hence AC equals m. All radii of a circle are equal by definition. (See Art. 25.) The side BC is a radius of the circle CEF, hence BC equals m. Explain why. The side AB was constructed equal to n. Hence the triangle CAB fulfils the given conditions, having two sides equal to m, and one side equal to n. EXERCISES 1. Do the two circles in the above construction necessarily intersect? Suppose the line n were more than twice as long as m, what then? Can a triangle be formed with sides 6 feet, 6 feet, and 13 feet in length? 2. If the line-segments m and n are equal, what sort of triangle does CAB become? 3. Construct a triangle two of whose sides shall be each equal to the longer line-segment n, and the third side equal to the shorter linesegment m. 4. If AF, BF, and CF are joined, show that ACF and BCF are each isosceles triangles. 5. Show how to find a point equidistant from two given points, making use only of the compasses. 38. For convenience, one side of a triangle is sometimes called the base of the triangle, and the other two, the sides. The base may be any side whatever; the opposite angle is then called the vertical angle, and the vertex of that angle, the vertex of the triangle. The two angles adjacent to the base are called the base angles. In an isosceles triangle it is usually the unequal side which is called the base. |