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Ex. 514. Given a point, A, between a circumference and a straight line. Through A, to draw a line terminated by the circumference and the given line, and bisected in A.

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Ex. 515. Given two points, A and B, on the same side of a line, CD. To find a point, X, in CD, such that AXC = LBXD.

Ex. 516. The bisectors of the angles of a circumscribed quadrilateral meet in a point.

BOOK III

PROPORTION. SIMILAR POLYGONS

255. A proportion is a statement expressing the equality of two ratios, as or a b = c: d.

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b

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d

256. The first and the fourth terms of a proportion are called the extremes, the second and the third, the means.

257. 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.

258. 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.

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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.

259. The last term is the fourth proportional to the first three. Thus, in the proportion, a : b = c: d, d is the fourth proportional to a, b, and c.

260. A series of equal ratios is called a continued proportion.

261. The two terms of a ratio must be either quantities of the same kind, or the quantities must be represented by their numerical measures only.

PROPOSITION I. THEOREM

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

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263. COR. If three terms of a proportion are respectively equal to the three corresponding 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 = 48.
Ex. 518. Find the value of x if a: m = x : n.

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.

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Ex. 519. If ab = mn, find all possible proportions consisting of a, b,

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PROPOSITION III. THEOREM

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

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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.

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268. COR. If a: bc: d, and a = kc, then bkd.

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Q.E.D.

PROPOSITION V. THEOREM

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.

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Ex. 522. Transform the proposition, m: x =p: q, so that x 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.

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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.

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