374. PROBLEM. E To construct a triangle equal to a given polygon. D Required to construct a triangle equal to the polygon ABCDE. Construction. Suppose the polygon has five vertices. Let B, C, D be three consecutive vertices. Draw BD, and through C draw a parallel to DB, cutting AB produced in F. Draw DF In a similar manner draw DG. Then GFD is the triangle required. Proof. 1. ▲ DBF = ▲ DBC, and ▲ DAGA DAE. Why? 2. .. ▲ DBF+^ DAB+ ^ DAG = ^ DBC+▲ DAB +ADAE. That is, AGFD = polygon ABCDE. Why? Let the student show how to extend this method so as to apply to a polygon with six, seven, etc., vertices. 1. To construct a right triangle equal to a given triangle. 2. To construct a triangle equal to a given triangle, and having one angle equal to a given angle. 3. To construct a right triangle equal to a given triangle, and having one leg equal to a given line segment. SUGGESTION: First construct a triangle equal to the given triangle, and having one side equal to the given segment, § 373; then apply Ex. 1. 4. To construct an isosceles triangle equal to a given triangle and having the base equal to a given line segment. 376. PROBLEM. To construct a square equal to a given rec Required to construct a square equal to rect. ABCD. Construction. Produce DC to E, making CE = CB. On DE as a diameter describe a semicircle, cutting BC produced in F. Then CF is a side of the required square. Proof. 1. Draw FD and FE. 2. DFE is a right angle and FC1 DE. Why? 3... the square on FC is equal to the rectangle whose adjacent sides are DC and CE, or CB. § 369. 377. PROBLem. To construct a square equal to the sum, or dif ference, of two given squares. The solution of this problem is left to the student. 1. To construct a rectangle equal to a given parallelogram. 2. To construct a parallelogram equal to a given parallelogram, and having one side equal to a given line segment. SUGGESTION: The construction is similar to that of § 373. 3. To construct a rectangle equal to a given parallelogram, and having one side equal to a given line segment. 4. To construct a parallelogram equal to a given parallelogram, and having an angle equal to a given angle. 5. To construct a parallelogram equal to a given triangle. 6. To construct a square equal to a given triangle. SUGGESTION: First construct a parallelogram equal to the given triangle ; then apply § 376. 7. To construct a square equal to a given polygon. 8. To construct a square equal to the sum of three or more given squares. 9. To construct a square equal to the sum of two or more given triangles. 10. To construct a right triangle equal to a given triangle, and having the hypotenuse equal to a given line segment. See § 325. 11. To construct a triangle equal to a given triangle, having one side equal to a given line segment, and the angle opposite that side equal to a given angle. See § 335. 12. To find a point within a triangle, such that the segments joining this point to the vertices shall divide the triangle into three equal parts. 13. To divide a given triangle into two equal parts by a line through a given point on one of the sides. 14. An attempt is made to row a boat at the rate of 4 miles an hour directly across a stream flowing at the rate of 3 miles an hour. If the stream is a mile wide, how long will it take to row across, and how far will the boat move? See Ex. 11, § 281. Ans. 15 minutes; 1 miles. 15. A sailor climbs a mast at the rate of 5 feet a second. The ship is sailing at the rate of 12 feet a second. Over what space does he actually move during 15 seconds? Ans. 195 feet. 16. A railway train moves south westward at the rate of 40 miles an hour. (a) How fast is it moving westward? (b) How fast is it moving southward? Ans. 28.28 miles an hour. 17. A man pushes at one corner of the rear end of a street car with a force of 70 pounds, and in a direction that makes an angle of 30° with the car's line of motion. How many pounds effective force does he exert in the direction of the line of motion of the car? Ans. 60.6 lb. CHAPTER VII PROPORTION. SIMILAR POLYGONS 379. In the preceding chapters we have studied the relations between figures which have the same size and shape (congruent figures), and between figures that have the same size but which differ in shape (equal figures). We wish now to investigate the properties of figures that have the same shape, but which differ in size (similar figures). Figures 175 and 176 are reproductions of the same portrait. Evidently they have the same shape, though they differ in size. Observe that each of the equal squares shown in Fig. 176 corresponds to one of the squares shown in Fig. 175. By actual measurement it will be found that the sides of the former are two-thirds the sides of the latter, while their areas are to each other as 4 to 9. From this example, it will be seen that the study of figures which have the same shape, but which differ in size, is intimately connected with ratio and proportion. 380. Terms of a ratio. In the ratio tio, a is called the antece dent and b is called the consequent; each is called a term of the ratio. 381. THEOREM. In a sequence of equal ratios, the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent. .*. a + c + e + g a+c+e+g= br+dr + fr+ hr = r(b + d +f+h). Why? =r= = etc. α с b d' Why? 382. A proportion. An equality of two ratios is called a proportion. The proportion, also written a:bc:d, or a:b::c:d, b is read a is to b as c is to d; a and b are also said to be in proportion to c and d. 383. Terms of a proportion. In the proportion a с each of b ď the numbers a, b, c, and d is called a term of the proportion. The first and last terms, a and d, are called the extremes; and the second and third terms, b and c, are called the means. 384. Fourth proportional, mean proportional, third proportional. The last term of a proportion is called a fourth proportional to the other three terms. α b If three numbers a, b, and c are in the proportion = then b is b с called the mean proportional between a and c; and c is called a third proportional to a and b. |