314. Area of a Rectangle. From the foregoing exercises it appears that the area of a rectangle in square units is equal to the product of its two dimensions in linear units, even if those dimensions are fractional, decimal, or irrational. RATIO 315. If two magnitudes of the same kind, such as two linesegments, contain a certain unit a and b times respectively, a b then the quotient is often called the ratio of these two magnitudes. 30, For example, the ratio of a line 10 ft. long to one 30 ft. long is 18, or. The ratio of m ton is often written m: n, or m÷ n. In the ratio a: b, a and b are called the terms of the ratio; a is called the antecedent and b the consequent. 316. Since ratios are really fractions, their properties are the properties of fractions. Hence The value of a ratio is not changed if both of its terms are multiplied or divided by the same number. 317. When two ratios a: band c:d are equal, the four numbers a, b, c, d, are said to be in proportion or to be proportional. This equality of ratios may be written in any one of the following forms: These are read " a is to bas c is to d.” 318. Since the ratio between two magnitudes of the same kind is obtained exactly or approximately as the quotient of their numerical measures, it is customary to extend the use of the term "ratio" to include the quotient of the numerical measures of two magnitudes of different kinds. Thus the number of pounds of pressure per square foot of area of a gas inclosed in a vessel is defined as the ratio of the total pressure to the total area of the inclosing vessel. 319. The ratio of two magnitudes of the same kind is independent of the unit by which the magnitudes are measured, since a change in the unit results merely in multiplying or dividing the two terms of the ratio by the same number. Thus the ratio of the areas of two rectangles is the same, whether they are both measured in square inches or in square centimeters. 320. The ratio of two magnitudes of different kinds is dependent upon the units by which the two magnitudes are measured. Thus the ratio of the weight of a physical solid in pounds to its volume in cubic feet is not the same as the ratio of its weight in grams to its volume in cubic centimeters. EXERCISES 1. Simplify the following ratios: 8:24; 31:7;41:47;.4:.03; (a + b)2: (a + b). 2. What is the ratio of a straight angle to a right angle? of the interior angle of a regular hexagon to the interior angle of an equilateral triangle? 3. Divide an angle of 180° into five parts in the ratio of 1:2:3:4:5. How many degrees in each angle? 4. Divide 50 in the ratio of m: n. 5. The sides of a triangle are in the ratio of 3:57. The perimeter of the triangle is 30. Find the length of each side. 6. Find the ratio of the areas of two rectangles if their respective dimensions in feet are 12 by 27 and 18 by 20; 3 by 7 and 23 by 9. 7. What is the ratio of the area of a given square to the area of the square on its diagonal? 8. What is the ratio of two rectangles, the first of base 10 and altitude x, and the second of base 12 and altitude x? 9. What is the ratio of two rectangles on the same base b, if their respective altitudes are 15 and 20? 10. What is the ratio of the area in square feet of a square, each of whose sides is 6 feet, to the length of one of its sides? Is the ratio the same if the measurements are taken in inches? in yards? AREAS OF SIMPLE FIGURES 321. Fundamental Principle. The area of a rectangle is equal to the product of its base and altitude. Thus if a and b are the altitude and base respectively of the rectangle 322. COROLLARY 1. Two rectangles are to each other as the products of their bases and altitudes. For if R = ab, and R' = a'b', then R ab R' a'b' 323. COROLLARY 2. Two rectangles having equal bases are to each other as their altitudes. 324. COROLLARY 3. Two rectangles having equal altitudes are to each other as their bases. 325. COROLLARY 4. Two rectangles having equal altitudes and equal bases are equal. The above corollaries may be written symbolically as follows: 326. COROLLARY 5. The area of a right triangle is equal to one half the product of its legs. B 327. COROLLARY 6. The area of a kite (rhombus, square, see § 151) is equal to one half the product of its diagonals. EXERCISES 1. The edge of a cube is 4 in. How many square feet in its entire exterior surface? 2. A map is drawn so that 1 cm. represents 1000 m. What area is represented by 1 sq. cm.? 3. The dimensions of a rectangular window are 5 ft. and 3 ft. It is to be divided into rectangles and squares as shown in the figure. Determine the dimensions of these parts. 4. The accompanying diagrams represent cross sections of steel beams. Determine the areas of the cross sections, using the dimensions given in millimeters in the following table: 5. The diagonals of a square are each 10 ft. long. Find the area of the square (§ 327). 6. The diagonals of a rhombus are 8 ft. and 7 ft. long. Find the area of the rhombus (§ 327). 7. If pand A represent the perimeter and the area respectively of a rectangle, determine the dimensions, x and y, in the exercises shown in the following table : 8. The Greek historian Thucydides (430 в.с.) estimated the size of the island of Sicily by means of the time it took to sail around it. What was the fallacy in his method? PROPOSITION I. THEOREM 328. The area of a parallelogram is equal to the product of Given the parallelogram ABCD, with its base AB equal to b, and its altitude BE equal to h. To prove that the area of the ABCD = bxh. Proof. 1. Draw the line A F || to BE, meeting CDproduced at F. 4. Then area ABED+△BCE = area ABED + ∆ADF. Ax. 2 That is, 5. But ABCD = rectangle ABEF. area rectangle ABEF=bh. ... area ABCD = bh. Why? 329. COROLLARY 1. Two parallelograms are to each other as the products of their bases and altitudes. 330. COROLLARY 2. Two parallelograms having equal bases are to each other as their altitudes. 331. COROLLARY 3. Two parallelograms having equal altitudes are to each other as their bases. 332. COROLLARY 4. Two parallelograms having equal bases and equal altitudes are equal. |