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268. Since a proportion may be regarded as an equation in which both members are fractions, it follows that:

1. The changes which may be made upon the members of an equation without destroying its equality may be made upon the couplets of a proportion without destroying the equality of the ratios.

2. The changes which may be made upon the terms of a fraction without altering the value of the fraction may be made upon the terms of each ratio of the proportion without destroying the propor

tion.

Proposition I

=

269. 1. Form several proportions, as 3:5 9:15, and discover how the product of the extremes compares with the product of the means in each.

2. If the means in any proportion are the same, how may the means be found from the product of the extremes?

3. Form a proportion whose consequents are equal. How do the antecedents compare?

4. Form a proportion in which either antecedent is equal to its consequent. How does the other antecedent compare with its consequent?

Theorem. In any proportion, the product of the extremes is equal to the product of the means.

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270. Cor. I. A mean proportional between two quantities is equal to the square root of their product.

If a:bb: c, find the value of b.

271. Cor. II. If in any proportion any antecedent is equal to its consequent, the other antecedent is equal to its consequent.

272. Cor. III. If the consequents of any proportion are equal, the antecedents are equal, and conversely.

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273. 1. If the product of the extremes of a proportion is 48, what may the extremes be? If 72? If 30? If 36? If 6a2? If 12 ab? If abc? 2. If the product of the means is 48, what may the means be? If 96? If 108? If 6 bcd? If abcd? If a2b2? If abc?

3. Form a proportion the product of whose extremes or means is 60; 72; 84; 80; 64; 144; x2y2, xyz; xyzv.

Theorem. If the product of two quantities is equal to the product of two others, either two may be made the extremes of a proportion of which the other two are the means.

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Ex. 393. If the vertical angle of an isosceles triangle is 30°, what is its ratio to each of the base angles?

Ex. 394. If the exterior angle at the base of an isosceles triangle is 100°, what is its ratio to each angle of the triangle ?

Ex. 395. If one of the acute angles of a right triangle is 40°, what is its ratio to the other acute angle? To the right angle? Ex. 396. The interior angles on the same two parallel lines are to each other as 3 to 2. in each angle?

side of a transversal cutting How many degrees are there

Ex. 397. The vertical angle of an isosceles triangle has the same ratio to a right angle that an angle of 40° has to an angle of an equilateral triangle. How many degrees are there in each angle of the isosceles triangle?

Proposition III

274. 1. Form a proportion and transpose the means. How do the resulting ratios compare?

2. Transpose the extremes. How do the resulting ratios compare? 3. Transform similarly and investigate other proportions.

Theorem. In any proportion, the first term is to the third as the second is to the fourth; that is, the terms are in proportion by alternation.

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275. 1. Form a proportion. If the antecedent of each ratio becomes the consequent, and the consequent the antecedent, how do the resulting ratios compare?

2. Transform similarly and investigate other proportions.

Theorem. In any proportion, the ratio of the second term to the first is equal to the ratio of the fourth term to the third; that is, the terms are in proportion by inversion.

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Proposition V

276. 1. Form a proportion. How does the ratio of the sum of the first two terms to either term compare with the ratio of the sum of the last two terms to the corresponding term?

2. Transform similarly and investigate other proportions.

Theorem. In any proportion, the ratio of the sum of the first two terms to either term is equal to the ratio of the sum of the last two terms to the corresponding term; that is, the terms are in proportion by composition.

Data:

To prove

a: b = c: d.

a+b: b = c +d: d, and a + b: a=c+d: c.

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In like manner it may be shown that a + b: a = c + d : c.

Therefore, etc.

Proposition VI

Q.E.D.

277. 1. Form a proportion. How does the ratio of the difference of the first two terms to either term compare with ratio of the difference of the last two terms to the corresponding term?

2. Transform similarly and investigate other proportions.

Theorem. In any proportion, the ratio of the difference between the first two terms to either term is equal to the ratio of the difference between the last two terms to the corresponding term; that is, the terms are in proportion by

division.

Data:

To prove

a: bc: d.

a − b : b = c — d: d, and ab: a=c-d: c.

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manner it may be shown that a ― b: a = c d : c.

re, etc.

14

Q.E.

Proposition VII

Form a proportion. How does the ratio of the sum of th erms to their difference compare with the ratio of the sum o terms to their difference?

em.

In any proportion, the ratio of the sum two terms to their difference is equal to the rat um of the last two terms to their difference; the rms are in proportion by composition and division

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