BOOK IV. AREAS OF POLYGONS. 392. DEF. The unit of surface is a square whose side is a unit of length. 393. DEF. The area of a surface is the number of units of surface it contains. 394. DEF. Plane figures that have equal areas but cannot be made to coincide are called equivalent. NOTE. In propositions relating to areas, the words "rectangle," "triangle," etc., are often used for "area of rectangle," "area of triangle," etc. PROPOSITION I. THEOREM. 395. Two rectangles having equal altitudes are to each other as their bases. Let the rectangles AC and AF have the same altitude AD. To that rect. AC: rect. AF prove = base AB: base AE. E CASE 1. When AB and AE are commensurable. Proof. Suppose AB and AE have a common measure, as AO, which is contained m times in AB and n times in AE. Apply 40 as a unit of measure to AB and AE, and at the several points of division erect Is. Proof. Divide AB into any number of equal parts, and apply one of them to AE as many times as AE will contain it. Since AB and AE are incommensurable, a certain number of these parts will extend from A to some point K, leaving a remainder KE less than one of the equal parts of AB. If the number of equal parts into which AB is divided is indefinitely increased, the varying values of these ratios will continue equal, and approach for their respective limits the ratios rect. AF and rect. AC AE (See § 287.) 396. COR. Two rectangles having equal bases are to each other as their altitudes. PROPOSITION II. THEOREM. 397. Two rectangles are to each other as the products of their bases by their altitudes. Let R and R' be two rectangles, having for their bases b and b', and for their altitudes a and a', respectively. Proof. Construct the rectangle S, with its base equal to that of R, and its altitude equal to that of R'. The products of the corresponding members of these equations give R R ахъ a' x b' Q. E. D. Ex. 349. Find the ratio of a rectangular lawn 72 yards by 49 yards to a grass turf 18 inches by 14 inches. Ex. 350. Find the ratio of a rectangular courtyard 18 yards by 151 yards to a flagstone 31 inches by 18 inches. Ex. 351. A square and a rectangle have the same perimeter, 100 yards. The length of the rectangle is 4 times its breadth. Compare their areas. Ex. 352. On a certain map the linear scale is 1 inch to 5 miles. How many acres are represented on this map by a square the perimeter of which is 1 inch? PROPOSITION III. THEOREM. 398. The area of a rectangle is equal to the product of its base by its altitude. Let R be a rectangle, b its base, and a its altitude. (two rectangles are to each other as the products of their bases and altitudes). 399. SCHOLIUM. When the base and altitude each contain the linear unit an integral number of times, this proposition. is rendered evident by dividing the figure into squares, each equal to the unit of surface. Thus, if the base contains seven linear units, and the altitude four, the figure may be divided into twenty-eight squares, each equal to the unit of surface. PROPOSITION IV. THEOREM. 400. The area of a parallelogram is equal to the product of its base by its altitude. Let AEFD be a parallelogram, b its base, and a its altitude. To prove that the area of the AEFD = a × b. Proof. From A draw AB || to DC to meet FE produced. Then the figure ABCD is a rectangle, with the same base and the same altitude as the AEFD. The rt. ABE and DCF are equal. § 151 § 178 From ABFD take the ▲ DCF; the rect. ABCD is left. From ABFD take the ▲ ABE; the □ AEFD is left. .. rect. ABCD ≈□ AEFD. Ax. 3 § 398 Ax. 1 Q. E. D. 401. COR. 1. Parallelograms having equal bases and equal altitudes are equivalent. 402. COR. 2. Parallelograms having equal bases are to each other as their altitudes; parallelograms having equal altitudes are to each other as their bases; any two parallelograms are to each other as the products of their bases by their altitudes. |