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PROPOSITION I. THEOREM
262. In any proportion, the product of the means is equal to the product of the extremes. Нур. .
263. Cor. If the first three terms of a proportion are respectively equal to the first three terms of another proportion, the fourth terms are also equal.
264. Note. — The product of two quantities, in Geometry, means the product of the numerical measures of the quantities.
Ex. 517. Find the value of x if 3: x = 4:8.
PROPOSITION II. THEOREM 265. 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. Hyp.
mn = pq
=Pq. Dividing both members by np
Ex. 519. If ab = mn, find all possible proportions consisting of a, b, m, and n.
266. A mean proportional between two quantities is equal to the square root of their product.
Extracting the square root of both members
Ex. 520. Find the mean proportional between 2 and 50, between a + m and a – m.
Ex. 521. Find the third proportional to m and n.
PROPOSITION IV. THEOREM
267. 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.
268. Cor. If a:b=c:d, and a = kc, then b=kd.
269. 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. Нур.
a:b=c:d. To prove
(262) .. b:a=d: c.
Ex. 522. Transform the proposition, m : x =p:9, so that a becomes the fourth term.
PROPOSITION VI. THEOREM
270. 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. Нур. .
PROPOSITION VII. THEOREM 271. If four quantities are in proportion, they are in proportion by division, i.e. the difference of the first two terms is to the second term as the difference between the last two terms is to the fourth term.
Ex. 523. If x +y:y= 7:3, find the ratio of x and y.
PROPOSITION VIII. THEOREM
272. If four quantities are in proportion, they are in proportion by composition and division, i.e. the sum of the first two terms is to their difference as the sum of the last two terms is to their difference.
Ex. 525. If x + y: x - y = 12:5, find the ratio of u and y.
PROPOSITION IX. THEOREM
273. In a continued proportion the sum of any number of antecedents is to the sum of the corresponding consequents as any antecedent is to its consequent.
ab = ab.
ad = bc.
af=be. By adding the equations
ab + ad + af = ab + bc + be,
a(b + d +1)=b(a +c+e). .. atcte: b + d +f=a:b.
Ex. 527. If a :b = c:d=e:f = 5:7, find a +c+e.
b + d + f
PROPOSITION X. THEOREM
274. The products of the corresponding terms of two or more proportions are in proportion.