5. The ALTITUDE OF A PARALLELOGRAM, is the perpendicular distance between two opposite sides. These sides are called bases; one the upper, and the other, the lower base. 6. The ALTITUDE OF A TRAPEZOID, is the perpendicular distance between its parallel sides. These sides are called bases; one the upper, and the other, the lower base. 7. The AREA OF A SURFACE, is its numerical value expressed in terms of some other surface taken as a unit. The unit adopted is a square described on the linear unit, as a side. PROPOSITION I. THEOREM. Parallelograms which have equal bases and equal altitudes, are equal. Let the parallelograms ABCD and EFGH have equal bases and equal altitudes: then will the parallelograms be equal. The triangles DAH and CBG, have the sides AD and BC equal, because they are opposite sides of the parallelogram AC (B. I., P. XXVIII.); the sides AH and BG equal, because they are opposite sides of the parallelogram AG; the angles DAH and CBG equal, because their sides are parallel and lie in the same direction (B. I., P. XXIV.): hence, the triangles are equal (B. I., P. V.). If from the quadrilateral ABGD, we take away the triangle DAH, there will remain the parallelogram AG; if from the same quadrilateral ABGD, we take away the tritriangle CBG, there will remain the parallelogram AC: hence, the parallelogram C is equal to the parallelogram EG (A. 3); which was to be proved. PROPOSITION II. THEOREM. A triangle is equal to one-half of a parallelogram having an equal base and an equal altitude. Let the triangle ABC, and the parallelogram ABFD, have equal bases and equal altitudes: then will the triangle be equal to one-half of the parallelogram. then, because they have equal altitudes, the vertex of the triangle will lie in the upper base of the parallelogram, or in the prolongation of that base. equal to From A, draw AE parallel to BC, forming the parallelogram ABCE This parallelogram will be the parallelogram ABFD, from Proposition I. But the triangle ABC is equal to half of the parallelogram ABCE (B. I., P. XXVIII., C. 1): hence, it is equal to half of the parallelogram ABFD (A. 7); which was to be proved. Cor. Triangles having equal bases and equal altitudes are equal, for they are halves of equal parallelograms. PROPOSITION III. THEOREM. Rectangles having equal altitudes, are proportional to their bases. There may be two cases: the bases may be commensurable, or they may be incommensurable. 1°. Let ABCD and HEFK, be two rectangles whose altitudes AD and HK are equal, and whose bases AB and HE are commensurable : then will the areas of the rectangles be proportional to their bases. HE, E Suppose that AB is to as 7 is to 4. Conceive AB to be divided into 7 equal parts, and HE into 4 equal parts, and at the points of division, let perpendiculars be drawn to AB and HE. Then will ABCD be divided into 7, and HEFK into 4 rectangles, all of which will be equal, because they have equal bases and equal altitudes (P. I.): hence, we have, ABCD : HEFK :: : 7 : 4. But we have, by hypothesis, AB : HE :: 7 : 4. From these proportions, we have (B. II., P. IV.), ABCD : HEFK :: AB : HE. Had any other numbers than 7 and 4 been used, the same proportion would have been found; which was to be proved. 2o. Let the bases of the rectangles be incommensurable : then will the rectangles be proportional to their bases. For, place the rectangle HEFK upon the rectangle ABCD, so that it shall take the position AEFD. Then, if the rectangles are not proportional to their bases, let us suppose that D F A ABCD : AEFD :: : AB : AO; ETOB Divide AO into in which AO is greater than AE. equal parts, each less than OE; at least one point of division, as I, will fall between E and 0; at this point, draw IK perpendicular to AB. Then, because AB and ΛΙ are commensurable, we shall have, from what has just been shown, ABCD : AIKD :: AB : AI. The above proportions have their antecedents the same in each; hence (B. II., P. IV., C.), The rectangle AEFD is less than AIKD; and if the above proportion were true, the line AO would be less than AI; whereas, it is greater. The fourth term of the proportion, therefore, cannot be greater than AE. In like manner, it may be shown that it cannot be less than AE; consequently, it must be equal to AE: hence, ABCD : AEFD :: AB AE; which was to be proved. : Cor. If rectangles have equal bases, they are to each other as their altitudes. PROPOSITION IV. THEOREM. Any two rectangles are to each other as the products of their bases and altitudes. H D Let ABCD and AEGF be two rectangles: then will ABCD be to AEGF, as AB × AD is to AE × AF. For, place the rectangles so that the angles DAB and EAF shall be opposite or vertical; then, produce the sides CD and GE till they meet in H. E A F AD: hence (P. III.), ABCD : ADHE :: AB : AE. The rectangles ADHE and AEGF have the same altitude AE: hence, ADHE : AEGF :: AD : AF. Multiplying these proportions, term by term (B. II., P. XII.), and omitting the common factor ADHE (B. II., P. VII.), we have, ABCD : AEGF :: AB × AD : AE× AF; which was to be proved. Scholium 1. If we suppose AE and AF, each to be equal to the linear unit, the rectangle AEGF will be the superficial unit, and we shall have, ABCD : 1 :: AB × AD : 1; |