Ex. 759. Given two points, A and B, on the same side of a line, CD. Tο find a point, X, in CD, such that ∠AXC = ∠ BXD. [See practical problems 44-53, pp. 293 and 294.] BOOK III PROPORTION. SIMILAR POLYGONS 270. DEF. A proportion is a statement of the equality of NOTE. The statements a = and a:b=c: d, are absolutely identical. Hence, if a hypothesis states a : b = c : d, we may in the proof employ this statement in the form without assigning a special reason. bd Similarly, if we have to prove a: b = c: d, we may prove instead 271. DEF. The first and the fourth terms of a proportion are called the extremes, the second and the third, the means. C bd 272. DEF. The first and the third terms are called the antecedents, the second and the fourth the consequents. Thus, in the proportion, a : b = c :d, a and d are the extremes, b and c the means, a and c the antecedents, and b and d the consequents. 273. DEF. When the means of a proportion are equal, either mean is said to be the mean proportional between the first and the last terms, and the last term is said to be the third proportional to the first and the second terms. Thus, in the proportion, a : b = b : c, b is the mean proportional between a and c, and cis the third proportional to a and b. 274. DEF. The last term is the fourth proportional to the first three. Thus, in the proportion, a : b = c :d, dis the fourth proportional to a, b, and c. 275. The two terms of a ratio must be either quantities of the same denomination, or the quantities must be represented by their numerical measures only. PROPOSITION I. THEOREM 276. In any proportion, the product of the means is equal to the product of the extremes. Clearing of fractions, i.e. multiplying both members by bd, 277. COR. If any three terms of a proportion are respectively equal to the three corresponding terms of another proportion, the remaining terms are equal. 278. NOTE. The product of two quantities, in Geometry, means the product of the numerical measures of the quantities. 279. If the product of two numbers is equal to the product of two other numbers, either two may be made the means, and the other two the extremes, of a proportion. Ex. 763. If ab = mn, find all possible proportions consisting of a, b, m, and n. Ex. 764. Form two proportions commencing with 3 from the equation 3 x 10 = 5 x 6. Ex. 765. If ab = xy, form two proportions commencing with b. Ex. 766. Find the ratio of x : y, if 280. A mean proportional between two quantities is equal to the square root of their product. PROPOSITION IV. THEOREM 281. If four quantities are in proportion, they are in proportion by alternation, i.e. the first term is to the third as the second is to the fourth. 282. If four quantities are in proportion, they are in proportion by inversion, i.e. the second term is to the first as the fourth is to the third. Ex. 768. Transform the proposition, m: x = p : q, so that x becomes the fourth term. PROPOSITION VI. THEOREM 283. If four quantities are in proportion, they are in proportion by composition, i.e. the sum of the first two terms is to the second term as the sum of the last two terms is to the fourth term. |