28. Since an angle inscribed in a semicire is angle, it follows from Arts. 26 and 27, that: tight $8 GEOMETRY 15 The leg AC is a mean proportional between A B and A That is PROPERTY OF THIS BOOK IS THE INTERNATIONAL TEXTBOOK COMPANY. (a) A perpendicular CD, Fig. 21, drawn from any point on the circumference of a circle to a diameter AB, is a mean proportional between the segments into which it divides the diameter; that is, AD: CD = CD: DB (b) A chord CA drawn from a point in a circumference to the end of a diameter is a mean proportional between the FIG. 21 whole diameter and the adjacent segment AD; that is, AB: AC = AC: AD 29. If from a point without a circle, a tangent and a secant are drawn, the tangent is a mean proportional between the whole secant and the exterior segment; that is, in Fig. 22, PB: PT = PT: PA. B FIG. 22 Ꭱ P In the triangles BPT and APT, the angle P is common. The angle B, an inscribed angle, and the angle PTA, an angle formed by a tangent and a chord, are equal, since each is measured by onehalf the same arc AT. Hence, the tri16, and angles are similar by Art. 30. If from a point without a circle any two secants are drawn, the product of one secant and its external segment is equal to the product of the other secant and its external segment. 31. If any two chords be drawn through a point within a circle, the product of the segments of one is equal to the product of the segments of the other. In Fig. 24, the angles D and B, being meas ured by one-half the arc A C, are equal. The Therefore, P 3 1. The perpendicular from the vertex of the right angle of a right triangle divides the hypotenuse into parts of 23.04 inches and 1.96 inches. Find: (a) the length of the perpendicular; (b) the length of the two sides of the triangle. (a) 6.72 in. Ans. {(8) 24 in. and 7 in. 2. If, in Fig. 22, the distance CP of the point P from the center of the circle is 65 feet, and the radius CR is 25 feet, what is the length of the tangent PT? Ans. 60 ft. 3. The chord of the arc of a segment is 14 inches long and the height of the segment is 2 inches; what is the radius? Ans. 13 in. OTHER SIMILAR POLYGONS 32. Two polygons are similar when they are composed. of the same number of triangles similar each to each and similarly placed. Thus, in Fig. 25, the polygons ABCDE and A' B' C' D' E' are composed of the same number of similar triangles similarly placed. Since the triangle A E D is similar to the triangle A' E' D', angle E angle E' and angle ADE = angle A' D' E'. Also, in the similar triangles ADC and = D angle Bangle B', and angle BAE triangles are similar, hence, ED: ED': = AD: A'D' and AD: A' D' = DC: D'C' = CB: C' B = BA: B'A' = AE: A' E' In like manner, DC: D'C' Therefore, as the angles of the one polygon are equal to the corresponding angles of the other and the sides of the one polygon are proportional to the sides of the other, the polygons are similar. 33. Two similar polygons can be divided into the same number of similar triangles similarly placed. 34. The perimeters of two similar polygons are in the same ratio as any two homologous sides. In Fig. 25, let P be the perimeter of the polygon A B C DE, and P' the perimeter of the polygon A'B'C' D' E'. Since the polygons are similar Let each of these equal ratios be denoted by R; that is, let RX A' E', ED = RX E' D', DC = R× D' C', Adding the sides of these equalities, AE+ED+DC+CB+B A = RX A' E' + RX E' D' + R × D'C' + R × C' B' + R × B'A' 35. Equation (1) of the preceding article is a series of equal ratios, of which the numerators are the antecedents and the denominators the consequents. The general truth was shown in that article, that in a series of equal ratios the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent. 36. Definitions.-The area of a surface is the superficial space included within its boundary lines. Area is expressed by the ratio of the surface to a surface of fixed value chosen as a unit and called the unit of area. 37. A square whose side is equal in length to the unit of length is usually taken as the unit of area, and its area is called the square unit. For example, if the unit of length is 1 inch, the unit of area, or square inch, is the square whose sides measure 1 inch, and the area of any surface is expressed by the number of square inches that the surface contains. If the unit of length were 1 foot, the unit of area would measure 1 foot on each side, and the area of the surface would be expressed in square feet. Square inch and square foot are abbreviated to sq. in. and sq. ft., respectively, and are often indicated by the symbols " and '. 38. Two surfaces are equivalent when their areas are equal. 39. Comparison of the Areas of Two Rectangles. The areas of two rectangles ABCD and A' B' C' D', Fig. 26, having equal altitudes are to each other as their bases; that is, area A B CD: area A' B' C' D' = AB: A' B'. = 5.4. Suppose that A' B' is four-fifths of A B, or that A B : A' B' equal parts A' E', E' F', etc. It is evident that A' E' = AE, for, since A B is to A'B' in the ratio of 5 to 4, any quantity, as A E, that is contained five times in AB must be contained four times in A' B'. Through the points of division E, F, E', F', etc., draw perpendiculars to AB and A'B'. Each large rectangle is thus divided into small rectangles, all of which are equal. As ABCD contains five, and A'B'C' D' contains four, of the small rectangles, the ratio of the two large rectangles is that of 5 to 4, which is also the ratio of their bases. 40. Since any of the sides of a rectangle can be considered as the base, it follows that the area of two rectangles having equal bases are to each other as their altitudes. 41. The areas of any two rectangles are to each other as the products of their bases by their altitudes. B a FIG. 27 Let A and B, Fig. 27, be two rectangles whose altitudes are a and a' and whose bases are b and b', respectively. Construct a rectangle C with an altitude a and a base b'. Then, by Arts. 39 and 40, Dividing the terms of the first member of equation (3) by C, A: Bab: a' b' |