Plane and solid geometry . al = trihedral Z Q, and with volumes denoted by Vand v\ respectively. „ V OA OC OB To prove —, = . V QF QG QM Argument 1. Place pyramid Q-FGM so that trihedral Z. Q shall coincide with trihedial Z pyramid Q-FGM in its newposition by 0-fgm. 2. From D and M draw DJ and mk ± plane OAC. A OAC DJ ^ ^, F A OAC DJ 3. Then — = r A OFG • mk A OFG MK 4. But A OAC OA . OC A ofg of . OG5. Again let the plane determined by DJ and3/^ intersect plane O^Cin line OKJ,Then rt. A DJO ~ rt. A _ ODMK~ OM^V OAOCV Reasons1. § 54, 14. 3. § 639. § 807. 6. 7. 8. .-. -1
Plane and solid geometry . al = trihedral Z Q, and with volumes denoted by Vand v\ respectively. „ V OA OC OB To prove —, = . V QF QG QM Argument 1. Place pyramid Q-FGM so that trihedral Z. Q shall coincide with trihedial Z pyramid Q-FGM in its newposition by 0-fgm. 2. From D and M draw DJ and mk ± plane OAC. A OAC DJ ^ ^, F A OAC DJ 3. Then — = r A OFG • mk A OFG MK 4. But A OAC OA . OC A ofg of . OG5. Again let the plane determined by DJ and3/^ intersect plane O^Cin line OKJ,Then rt. A DJO ~ rt. A _ ODMK~ OM^V OAOCV Reasons1. § 54, 14. 3. § 639. § 807. 6. 7. 8. .-. -1 = OD OAOCOD OF OG OM QF-QG QM 4. § 498. 5. §§613,616. 6. § 422. 7. § 424, 2. 8. § 309. 811. Def. Two polyhedrons are similar if they, have thesame number of faces similar each toeacli and similarly ])laced,and have tlieir corresponding polyhedral angles equal. BOOK VII 377 Proposition XV. Theorem 812. The volumes of two similar tetrahedrons are toeach other as the cubes of any two homologous 0 1^ Q A F Given similar tetrahedrons 0-ACD and Q-i^^G^if with volumesdenoted by V and F, and with OA and QF two homologousedges. To prove -- = ^^. Argument Eeasons 1. Trihedral Z. 0 =z trihedral Z Q. 1. §811. 9 , V _ OA OC OD _0A OC OB V^~ QF QG- QM~ QF QG QM 2. § 810. 3. But^^=^^=^^,QF QG QM 3. § 424, 2. M V OA OA OA OJ? 4. .•.—- = — . = -. F QF QF QF n^ 4. § 309. 813. Question. Compare §§ 810 and 812 with §§ 498 and 503. Are the same general methods used in the two sets of theorems ? 814. Note. The proposition, two similar convex polyhedrons areto each other as the cubes of any two homologous edges, will be assumedat this point, and will be applied in some of the exercises that follow. Fora complete discussion of this principle see Appendix, §§ 1022-1029. Ex. 1332. The edges of two regular tetrahedrons are 6 centimetersand 8 centimeters, respectively. Find the ratio of their volumes. Ex. 1333. The volumes of two similar polyhedrons
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometr, bookyear1912
