BOOK III PROPORTION AND SIMILAR FIGURES 382. Def. A proportion is the expression of the equality of two ratios. a b EXAMPLE. If the ratio is equal to the ratio, then the equation is a proportion. This proportion may also be written a : b = c d or abcd, and is read a is to b as c is to d. 383. Def. The four numbers a, b, c, d are called the terms of the proportion. 384. Def. The first term of a ratio is called its antecedent and the second term its consequent; therefore : The first and third terms of a proportion are called antecedents, and the second and fourth terms, consequents. 385. Def. The second and third terms of a proportion are called its means, and the first and fourth terms, its extremes. 386. Def. If the two means of a proportion are equal, this common mean is called the mean proportional between the two extremes, and the last term of the proportion is called the third proportional to the first and second terms taken in order; thus, in the proportion a: b=b: c, b is the mean proportional between a and c, and c is the third proportional to a and b. 387. Def. The fourth proportional to three given numbers is the fourth term of a proportion the first three terms of which are the three given numbers taken in order; thus, if a: b = c : d, d is called the fourth proportional to a, b, and c. PROPOSITION I. THEOREM 388. If four numbers are in proportion, the product of the extremes is equal to the product of the means. 389. Note. The student should observe that the process used here is merely the algebraic "clearing of fractions," and that as fractional equations in algebra are usually simplified by this process, so, also, proportions may be simplified by placing the product of the means equal to the product of the extremes. 390. Cor. I. The mean proportional between two numbers is equal to the square root of their product. 391. Cor. II. If two proportions have any three terms of one equal respectively to the three corresponding terms of the other, then the remaining term of the first is equal to the remaining term of the second. Ex. 606. Given the equation m: r = d: c; solve (1) for d, (2) for r, (3) for m, (4) for c. Ex. 607. Find the fourth proportional to 4, 6, and 10; to 4, 10, and 6; to 10, 6, and 4. Ex. 608. Find the mean proportional between 9 and 144; between 144 and 9. Ex. 609. Find the third proportional to and ; to 3 and . b. 392. Questions. What rearrangement of numbers can be made in Ex. 607 without affecting the required term? in Ex. 608? in Ex. 609 ? Ex. 610. Find the third proportional to a2 b2 and a Find the fourth proportional to a2 - b2, a *See § 382 for the three ways of writing a proportion. Ex. 611. b, and a + b. Ex. 612. If in any proportion the antecedents are equal, then the consequents are equal and conversely. 393. If the product of two numbers is equal to the product of two other numbers, either pair may be made the means and the other pair the extremes of a proportion. The proof that a and d may be made the means and b and c the extremes is left as an exercise for the student. 394. Note. The pupil should observe that the divisor in Arg. 2 above must be chosen so as to give the desired quotient in the first member of the equation: thus, if hl = kf, and we wish to prove that h f k we Ex. 617. Ex. 618. portions: Ex. 619. Given pt: = cr. Prove prc:t; also, c: p = t : r. r: ml:s, s : m = l : r, l:s=r:m, m:s=r: l. Form a proportion from 7 x 4 = 3×a; from ft = gb. How can the proportions obtained be verified? Ex. 620. Ex. 623. Form a proportion from (a + c) (a — b) = de. Form a proportion from (a + b) (a — b) = 4x, making x (1) an extreme; (2) a mean. Ex. 624. If 7x+3y: 12=2x+y: 3, find the ratio x:y. 395. If four numbers are in proportion, they are in proportion by inversion; that is, the second term is to the first as the fourth is to the third. 396. If four numbers are in proportion, they are in proportion by alternation; that is, the first term is to the third as the second is to the fourth. Ex. 625. If xy, what is the value of the ratio x: y? Transform a : x = 4:3 so that x shall occupy in turn every place in the proportion. 397. Many transformations may be easily brought about by the following method: (1) Reduce the conclusion to an equation in its simplest form (§ 388), then from this derive the hypothesis. (2) Begin with the hypothesis and reverse the steps of (1). This method is illustrated in the analysis of Prop. V. PROPOSITION V. THEOREM 398. If four numbers are in proportion, they are in proportion by composition; that is, the sum of the first two terms is to the first (or second) term as the sum of the last two terms is to the third (or fourth) term. Given a:bc: d. To prove: (a) a+b: a=c+d: c; (b) a+b:b=c+d: d. I. Analysis (1) The conclusions required above, when reduced to equations in their simplest forms, are as follows: (2) Now begin with the hypothesis and reverse the steps. (b) The proof of (b) is left as an exercise for the student. |