9. Given a rectangle of sides a and b, b>a. Divide the side b into two parts such that the difference of the squares on the parts shall be equivalent to the area of the rectangle. Ans. b+a b-a 2 2 Explain how this problem could be solved geometrically, that is, constructed by means of the compasses and straightedge. 10. Given three points A, B, and C in a straight line. Determine on the same straight line a point P such that, AP2=BP CP. Discuss fully. 11. Divide a line 20 in. long into three parts such that the first is to the second as 2:3, and the second is to the third as 2:5. Ans. 3 in.; 4 in.; 12 in. 12. Divide a line 16 in. long into two parts such that the square on one of the parts shall be equivalent to the product of the whole line and the other part. Ans. 9.89 in.; 6.11+ in. 13. Find the dimensions of a rectangle that is equivalent to a square 10 in. on a side, and has a perimeter double that of the square. Ans. 37.32 in. by 2.68 in. 14. Three points A, B, and C are located as shown in the figure. Three discs are centered at these points, each tangent to the other two. Find the diameters of the disks. Ans. At A, 0.8960 in.; at B, 0.9716 in.; at C, 0.9700 in. QUESTIONS 1. Which of the following figures have areas of fixed values if the perimeters are given: square, rectangle, right triangle, right isosceles triangle, equilateral triangle, rhombus, trapezoid, parallelogram with one angle 60°, isosceles trapezoid with one angle 60°, rhombus with angle 60°? 2. Of how many of the preceding can you determine the areas? 3. What measurements are necessary in order that one may determine the area of a square? A rectangle? A rhombus? A parallelogram? A triangle? An equilateral triangle? A trapezoid? A quadrilateral? A hexagon that is not regular? abcd, the four numbers a, b, c, and d are called the terms of the proportion. The terms a and d are called the extremes, and the terms b and c are called the means. 394. Fourth Proportional. The fourth term of a proportion is called the fourth proportional to the first three terms taken in order. α C In the proportion d is the fourth proportional to a, b, and c. b ď 395. Mean Proportional. If the two means of a proportion are equal, this common mean is called the mean proportional between the two extremes. 396. Continued Proportion. The numbers a, b, c, d, e, are 397. The terms of one ratio are said to be inversely proportional to the terms of another ratio when the first ratio equals the reciprocal* of the second. Thus, a and b are inversely proportional to c and d if α d b For example, 3 and 5 are inversely proportional to 15 and 9, for=1. *The reciprocal of a number is 1 divided by that number. Then, the reciprocal of a fraction is the fraction inverted. Thus the reciprocal of 3 is; the reciprocal of § is 1÷&=. FUNDAMENTAL THEOREMS 398. Theorem. If four numbers form a proportion, the product of the extremes equals the product of the means. Proof. Multiplying both ratios by bd, ad=bc. NOTE. A proportion is an equation and may be treated as such. 399. Theorem. The mean proportional between two numbers equals the square root of their product. 400. Theorem. If the two antecedents of a proportion are equal, the two consequents are equal. 401. Theorem. If three terms of one proportion are equal respectively to three corresponding terms of another proportion, the fourth terms are equal. 6. If x is a mean proportional between a and b, show that a : :b=a2: x2. 7. Find the fourth proportional to 1, 2, and 3. 8. Find the mean proportional between each of the following pairs of numbers: (1) 5 and 20, (2) 4 and 100, (3) 2xy and 8xy, (4) a+b and a−b. 402. Theorem. If the product of two numbers equals the product of two other numbers, either two may be made the means and the other two the extremes of a proportion. The student should note the number of proportions that may be formed and should obtain several of them. EXERCISES 1. Form all proportions possible from 5×6=3×10. 2. Form four proportions from 6=ax, using x as a mean. (1) a2-b2cd. (2) x2-1-a2 —b2. (3) ax+bx+cx=nd+nh+nk. 403. Theorem. In a series of equal ratios, the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent. THEOREMS OF TRANSFORMATION 404. Theorem. If four numbers form a proportion, they are in proportion by inversion; that is, the second term is to the first as the fourth is to the third. b d Making b and c the extremes, a and d the means, $ 402 a 405. Theorem. If four numbers form a proportion, they are in proportion by alternation; that is, the first term is to the third as the second is to the fourth. 810 81 с ad=bc. § 398 § 402 406. Theorem. If four numbers form a proportion, they are in proportion by addition; that is, the sum of the first two terms is to the second as the sum of the last two terms is to the fourth. |