Order of compounding. 27. THEOREM 4. The order in which two given ratios are compounded is indifferent. Let there be two ratios A: B, X: Y. To prove that the ratio obtained by compounding them is the same in whichever order they be taken. Take any line L. M to N in the ratio Alter L to M in the ratio A: B. Alter Then the two ratios (taken in the order named) have the auxiliary magnitudes L, M, N, and hence compound into the ratio L : N. Next take the two ratios in the order X: Y, A:B. Take any line L'. Alter L' to M' in the ratio X: Y. Alter M' to N' in the ratio A: B. Then the two ratios (taken in this order) have the auxiliary magnitudes L', M', N', and hence compound into the ratio L': N'. Now it is to be proved that L:N= L': N'. Find P a fourth proportional such that L: M = N:P. and M, N, P, L', M', N', have their successive ratios respectively equal, for and i.e. L': M' =X: Y=M: N, M': N' A: B=L: M = N:P. Hence the extremes of the two sets are proportional, now therefore M: P=L:N, = Hence the theorem is proved. Ex. The order of compounding three ratios is indifferent. [IV. 19 Equal ratios compounded with unequal ratios. 28. THEOREM 5. If one ratio is greater than another, then the ratio compounded of the greater and any third ratio is greater than that compounded of the less and the same third ratio. Take any line L. P: Q and X: Y. Alter it to M in the ratio X: Y. Alter M to N in the ratio A: B. Then the ratios X: Y and A: B have the auxiliary magni tudes L, M, N. Again alter M to N' in the ratio P : Q. [22, def. Then the ratios X: Y and P: Q have the auxiliary magni then the ratio compounded of A: B and L: M is greater than that compounded of P: Q and X : Y. [Apply the theorem twice.] Duplication of a ratio. 30. Definition. When two ratios are equal, the ratio compounded of them is called the duplicate of either of them. When three ratios are equal, the ratio compounded of them is called the triplicate of any one of the original ratios. 31. THEOREM 6. If three magnitudes are proportional, then the ratio of the first to the third is equal to the duplicate of the ratio of the first to the second. Given A: BB: C; to prove 4: C A: duplicate of 4: B. The ratio of 4: C is compounded of the successive ratios A: B and B C (22, def.). But these two ratios are equal. Therefore the ratio compounded of them is the duplicate of either (30, def.). Hence 4: C equals the duplicate of 4: B. 32. Cor. I. If four magnitudes are in continued proportion, the ratio of the first to the fourth is equal to the triplicate of the ratio of the first to the second. 33. Cor. 2. To find a ratio equal to the duplicate of a given line-ratio; also of a given numerical ratio. 34. Cor. 3. To find the triplicate of a given line-ratio. Comparison of duplicate ratios. 35. THEOREM 7. According as one ratio is greater than, equal to, or less than another, so is the duplicate of the former greater than, equal to, or less than the duplicate of the latter. the ratio compounded of A: B and ▲ : B is equal to the ratio compounded of P: Q and P: Q (23). the ratio compounded of A: B and A: B is greater than that compounded of P: Q and P: Q (29). Hence the theorem is established. 36. Cor. One ratio is greater than, equal to, or less than another according as the duplicate of the first ratio is greater than, equal to, or less than the duplicate of the second. SIMILAR TRIANGLES This section and the next will treat of similar triangles and similar polygons, respectively. In the following general definitions the word " polygon" will be understood to include "triangle." A former definition is here repeated for con`venience. 37. Definitions. Two polygons are said to be mutually equiangular if the angles of one, taken in order, are equal respectively to those of the other taken in order. The equal angles are said to correspond; and the sides joining the vertices of corresponding angles are called corresponding sides. Two polygons are said to have their sides proportional if the sides of one, taken in order as antecedents, form a series of equal ratios with the sides of the other taken in order as consequents. Two polygons are said to be similar if they are mutually equiangular, and if the corresponding sides are proportional. The ratio of any two corresponding sides is called the ratio of similitude of the similar polygons. E.g., the quadrangles ABCD and A'B'C'D' are similar if the angles A, B, C, D are equal respectively to A', B', C', D', and if AB: A'B' = BC: B'C' =CD :C'D' = DA: D'A'. Each of these ratios is equal to the ratio of similitude of the similar quadrangles. Two similar polygons are said to be directly or obversely similar according as they are directly or obversely equiangular (I. 187). Ex. Two regular polygons of the same number of sides are similar. CONDITIONS OF SIMILARITY The next four theorems relate to the conditions of similarity of two triangles. Angles equal. 38. THEOREM 8. If two triangles are mutually equiangular, then their sides are proportional; and the triangles are similar. Let the triangles ABC and A'B'C' be equiangular. 44 Α' To prove that Β' Α B" B AB: A'B' = BC: B'C' = CA : C'A'. Apply the triangle A'B'C' to ABC so that A' coincides with 4, and A'B' falls on AB; then A'c' falls on AC, because the angles 4 and A' are equal. Let B' and ' take the respective positions B" and c" on the sides AB and AC or else on their prolongations. Since the angles B and B" are equal, the lines BC and B"C" are parallel; therefore, by theorem 1, i.e. AB: AB' = AC: AC", AB: A'B' = AC: A'C'. Similarly by applying the angle B' to the angle B it may be shown that AB: A'B' = BC : B'C'. 39. Cor. I. A parallel to one side of a triangle forms with the other two sides a similar triangle. 40. Cor. 2. Triangles whose sides are parallel, respectively, are similar. |