6.5 Geometric Constructions
1.
a. Given the line through A and B and the point C, locate D on AB such that |AD| = |AC| . Then locate E such that |CE| = |DE|. The line through C and E is parallel to AB (ACED is a parallelogram).
2.
a. The line through (a, b) and (c, d) has equation 
c. Given ax + by = c and dx + ey = f in F[x], the solution (if any) involves only field operations in F, and so both x and y are in F (if they exist).
3. The question asks whether 
can be constructed. This is the result of bisecting 
. But the angle π/6 is constructible because 
is constructible (see diagram) so angle π/12 is constructible.
can be constructed. This is the result of bisecting . But the angle π/6 is constructible because is constructible (see diagram) so angle π/12 is constructible.
5. No. A sphere of radius 1 has volume 
and, if this is the volume of a cube with side a, then . But a is not constructible since it is not even algebraic over . For if ...
and, if this is the volume of a cube with side a, then . But a is not constructible since it is not even algebraic over . For if ...Get Introduction to Abstract Algebra, Solutions Manual, 4th Edition now with the O’Reilly learning platform.
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