.. truncated prism AM ~ truncated prism A'M'. (573) (578) (Ax. 3.) Q. E. D. 582. COR. Prisms having congruent right sections and equal lateral edges are equivalent. PROPOSITION V. THEOREM 583. The opposite lateral faces of a parallelopiped are congruent and parallel. A B' Given AB' and DC', opposite faces of parallelopiped AC'. B (568, 141) (495) (150) (491) Q. E. D. 584. COR. Any two opposite faces of a parallelopiped may be taken as bases. Ex. 1501. In a rectangular parallelopiped, lettered as the preceding figure, find the length of the diagonal AC', if AB = 9, BC = 12, CC′ = 8. Ex. 1502. In a right parallelopiped lettered similarly, find the length of the diagonal AC', if AB = 5, BC = 3, CC' = 2, and Z ABC = 120°. Ex. 1503. Find the total area of a right prism if its altitude is 12 in., and its base is a triangle the sides of which are 13 in., 15 in., and 4 in. respectively. Ex. 1504. Find the total area of a right prism if its altitude is 9 in., and its base is a rhombus of side 4 in. and having an acute angle of 60°. PROPOSITION VI. THEOREM 585. Two rectangular parallelopipeds having equal ses are to each other as their altitudes. Given AB and CD, the altitudes of parallelopipeds м and м' which have equal bases. I.e. if CD is divided into n equal parts, and one of them is laid off on AB, AB contains m such parts. Through the points of division pass planes parallel to the bases. [To be completed by the student.] 586. DEF. The dimensions of a rectangular parallelopiped are the three edges that meet at the same vertex. 587. COR. The preceding theorem may be stated: Two rectangular parallelopipeds which have two dimensions in common are to each other as the third dimension. PROPOSITION VII. THEOREM 588. Two rectangular parallelopipeds having equal altitudes are to each other as their bases. Given a, b, c and a, b', c', the three dimensions of rectangular parallelopipeds M and N respectively; a being the equal altitude. 589. COR. Two rectangular. parallelopipeds that have one dimension in common are to each other as the products of their other two dimensions. 590. NOTE. The product of three lines, or the product of an area and a line, denotes the product of the numerical measures of these quantities. Ex. 1505. Two rectangular parallelopipeds have equal altitudes and bases whose dimensions are 4 and 7, and 5 and 9 respectively. Find the ratio of their volumes. Ex. 1506. What is the ratio of the volumes of two rectangular parallelopipeds having equal bases and altitudes a and b respectively? PROPOSITION VIII. THEOREM 591. Two rectangular parallelopipeds are to each other as the products of their bases and their altitudes. Given M and N', two rectangular parallelopipeds, B and B' their bases, and a and a' their altitudes respectively. 592. COR. Two rectangular parallelopipeds are to each other as the products of their three dimensions. PROPOSITION IX. THEOREM 593. The volume of a rectangular parallelopiped is equal to the product of its three dimensions. M a Given a, b, c, the dimensions of a rectangular parallelopiped м. To prove HINT. volume of M = axbx c. Construct V unit of volume, and apply Prop. VIII. 594. COR. 1. The volume of a cube equals the cube of its edge. 595. COR. 2. The volume of a rectangular parallelopiped equals the product of its altitude by its base. |