Solid Geometry, With Problems And Applications
Solid Geometry, With Problems And Applications
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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.
What to read after Solid Geometry, With Problems And Applications?
You can find similar books in the "Read Also" column, or choose other free books by H E Herbert Ellsworth Slaught to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
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.
What to read after Solid Geometry, With Problems And Applications?
You can find similar books in the "Read Also" column, or choose other free books by H E Herbert Ellsworth Slaught to read onlineMoreLess
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