RECAPITULATION OF THE TRUTHS CONTAINED IN PART II. 1. On Proportions. Ques. 1. How is a geometrical ratio determined? Q. 2. What is the ratio of a line 3 inches in length to a line of 12 inches? What the ratio of a line 2 inches in length to one of 10 inches? &c. Q. 3. When two geometrical ratios are equal to one another, what do they form? Q. 4. What is a geometrical proportion? Q. 5. What signs are used to express a geometrical proportion? Q. 6. What sign is put between the two terms of a ratio? Q. 7. What sign is put between the two ratios of a proportion? Q. 8. What are the first and fourth terms of a geometrical proportion called? Q. 9. What are the second and third terms of a geometrical proportion called? Q. 10. What are the most remarkable properties of geometrical proportions? Ans. a. In every geometrical proportion the two ratios may be inverted. b. In every geometrical proportion the order of the means or extremes may be inverted. c. If two geometrical proportions have a ratio common, the two remaining ratios make again a proportion. d. If you have several geometrical proportions, of which the second has a ratio common with the first, the third a ratio common with the second, the fourth a ratio common with the third, &c., the sum of all the first terms will be in the same ratio to the sum of all the second terms as the sum of all the third terms is to the sum of all the fourth terms; that is, the sums make again a proportion. e. The second term of a proportion being added once, or any number of times, to the first term, and the fourth term the same number of times to the third term, they will still be in proportion; and in the same manner can the first term be added a number of times to the second term and the third the same number of times to the fourth term, without destroying the proportion. f. From three terms of a geometrical proportion the fourth term can be found. g. If four lines are together in a geometrical proportion, their lengths expressed in numbers of rods, feet, or inches, &c. will be in the same proportion. h. In every geometrical proportion the product obtained by multiplying the two mean terms together is equal to the product obtained by multiplying the two extreme terms together. Quest. How can you prove each of these principles ? * * QUESTIONS ON SIMILARITY OF TRIANGLES. Quest. What other principles do you recollect to have learned in the second part of the 2d section? Ans. 1. If one side of a triangle is divided into any number of equal parts, and then, from the points of division, lines are drawn parallel to one of the two other sides, the side opposite to the one that has been divided will by these parallels be divided into as many equal parts as the first side. 2. If, in a triangle, a line is drawn parallel to one of the sides, that parallel divides the two other sides into such parts as are in proportion to each other and to the whole of the two sides themselves; and the reverse of this principle is also true; namely, a line must be parallel to one of the sides of a triangle, if it divides the two other sides proportionally. 3. If, in a triangle, a line is drawn parallel to one of the sides, the triangle which is cut off by it, is similar to the whole triangle. 4. If the three angles in one triangle are equal to the three angles in another triangle, each to each, the two triangles are similar to one another; and the same is the case if only two angles in one triangle are equal to two angles in another, each to each. 5. If an angle in one triangle is equal to an angle in another, and the two sides which include that angle in the one triangle are in proportion to the two sides which include the equal angle in the other triangle, these two triangles are similar to each other. 6. If the three sides of a triangle are in proportion to the three sides of another triangle, these two triangles are similar to each other. * 7. If, in a right-angular triangle, a perpendicular is dropped from the vertex of the right angle upon the hypothenuse, that perpendicular divides the whole of the triangle into two triangles, each of which is similar to the whole triangle, and which are consequently similar to each other. 8. The perpendicular dropped from the vertex of a right-angular triangle to the hypothenuse is a mean proportional between the parts into which it divides the hypothenuse. 9. In every right-angular triangle, each of the sides which include the right angle is a * The teacher will do well to let the pupil repeat the different cases where two triangles are similar to each other. (page 87.) |