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ANSWERS TO GSP5 CONSTRUCTING PERPENDICULAR BISECTORS MANUAL

This entails constructing a side of the larger cube, and in this case that means constructing a length equal to the cube root of 2. A particular instance of this problem would be to construct a cube whose volume is twice that of the unit cube. The problem is to construct a cube that has twice the volume of a given cube.

The Delian problem - duplicating the cube.

Theorem: If a constructible number is a root of a cubic equation with rational coefficients, then the equation must have at least one rational root. The points could also be collinear.Īn algebraic analysis of the fields of constructible numbers shows the following: If N = A' there will be no nine-point circle, but N could equal O, or A' could equal O and the construction will still work. This construction always gives a unique triangle provided one exists. If the parallel is tangent to the circle there is one solution, and finally, if the parallel intersects the circle twice, there are two solutions (as indicated in the situation drawn above).Įxample: Construct a triangle, given the circumcenter O, the center of the nine-point circle N, and the midpoint of one side A'. If the parallel does not intersect the circle, there is no solution. Finally, if b is greater than c (the situation drawn above) then there will be four points of intersection.Įxample: Construct a triangle, given one angle, the length of the side opposite this angle, and the length of the altitude to that side.Īs the position of vertex A is determined by the intersection of a single line with a circle, there are three possibilities for the number of solutions.

If b equals c, the lines will be tangent to the circle and we would get two solutions. If b is less than c, there will be no intersection, so no solutions. Example: Construct a triangle, given the length of one side of the triangle, and the lengths of the altitude and median to that side.Īs the third vertex is determined by the intersection of one of two parallel lines with a circle, there are three possibilities for the number of solutions. Given two constructible numbers one can with straightedge and compass construct their:Ĭonstructions using basic operations. Note that the restrictions imply that the constructible numbers are limited to lying in certain quadratic extensions of the rationals. Given a segment which represents the number 1 (a unit segment), the segments which can be constructed from this one by use of compass and straightedge represent numbers called Constructible Numbers. Divide a segment into a given ratio (internal and external).

Partition a segment into n congruent segments.

Construct a parallel to a line through a given point.

Construct a perpendicular to a line from a point not on the line.

Construct a perpendicular to a line at a point on the line.

This underscores the statement above technique is not as important as understanding why it works. Of course, this means that how certain constructions were done by the ancient Greeks are quite different from the way we would do them today. The ancient dividers can do everything that modern compasses can.

Given the difference in the two tools, it appears that the modern compass is a more powerful instrument, capable of doing more things. Modern compasses remain open when picked up, so such transfers are possible. Should you pick up a divider, the arms will collapse, so it is impossible to use them to transfer lengths from one area to another. Dividers consist of just two arms with a central pivot. Rather, they used a device known as a divider. The ancient greek tool used to construct circles is not the modern day compass.

ANSWERS TO GSP5 CONSTRUCTING PERPENDICULAR BISECTORS MANUAL

It is not important to gain the manual dexterity needed to carry out a careful construction. It is important that one be able to analyze a construction to see why it works. The value of studying these constructions lies in the rich supply of problems that can be posed in this way. In other words, this is not a practical subject, if one is interested in constructing a geometrical object there is no reason to limit oneself as to which tools to use. However, the stipulation that these be the only tools used in a construction is artificial and only has meaning if one views the process of construction as an application of logic. Their use reflects the basic axioms of this system. Lecture Notes 5 - Math 3210 Lecture Notes 5 The philosophy of constructionsĬonstructions using compass and straightedge have a long history in Euclidean geometry.