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Why is there a need to use a straightedge and compass to construct geometric figures?
The compass is used to measure angles. The straightedge is used to draw a straight line. The two items together, are used to measure and draw angles and lines in geometric drawings.
Is there a need for students to understand how to use a compass and straightedge?
Very much so in certain aspects of geometry.
What is the difference between a straightedge and a ruler?
a straightedge need not have measuring increments (inches, centimeters, etc.) while a ruler is a straightedge with measuring increments
Trisecting a line segment by using only a straightedge and compass was proven impossible by advanced algebra.?
You don't need advanced algebra to prove that it is impossible to trisect a line segment using only a straight edge and a compass: anyone knows that you will also need a pencil! And one you have that then there are plenty of easy ways to do it.
Is it possible to construct a perpendicular bisector to any given line segment using only a straightedge and a compass?
Yes, but you will also need a pen or pencil. Also, if the line segment is very long (eg 5 metres) you may have difficulty in finding a compass that will do the trick!
