Solid Geometry, With Problems And Applications

Cover Solid Geometry, With Problems And Applications
Solid Geometry, With Problems And Applications
H E Herbert Ellsworth Slaught
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Theorem. The volumes of two tetrahedrons, having a trihedral angle of the one congruent to a trihedral angle of the other, are in the same ratio as the products of the edges which meet in the vertices of these angles.
Given the tetrahedrons P-ABC and P'-A'&C' whose volumes are Fand V and in which Tri. Zi>=Tri. ZP'.
To prove that PA ' PB ' PC Proof : Place P^-A^B^& so that Tri. Zp' coincides with Tri. Zp'.
Let CM and dM^ be the altitudes of P^ABC emdP-A^B'c^ from the vertices C and c' upon the p
...lane PAB.
Let AN and A'N^he the altitudes of the A PAB and PA'b'.
Then ^ _■ ^C3f*areaP^g _ CM PB - AN F' \ C'M^ . area PA^B' c'm' • PB^ • A^N^ (1) Now prove CM PC and ^y ^ PA &m' P& A'N^ Hence, substituting in (1), y^ PC' PB ' PA V' we have P^C' • p'b' • P'^' Give all the steps and reasons in detail.
PYRAMIDS AND CONJES. 79 147. EXERCISES.
1. Show that the lateral faces of a frustum of a regular pyramid are congruent isosceles trapezoids. Hence find the area of its lateral surface in terms of the slant height and the perimeters of the bases.


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