Constructing a line tangent to a circle

The  exercise appears under the Geometry Math Mission. This exercise uses compass and straightedge to create tangent lines to a circle through a point.

Types of Problems
There are two types of problems in this exercise:


 * 1) Find the tangent through the point on the circle: This problem provides a circle and a point on the circle. The student is asked to construct the tangent line to the circle through the point using only a compass and straightedge.Calttac1.png
 * 2) Find the tangent through the point off the circle: This problem provides a circle and a point off the circle. The student is asked to construct the tangent line to the circle through the point using only a compass and straightedge.Calttac2.png

Strategies
Comfort with constructions involving compass and straightedge would increase likelihood of success on this exercise. When first introduced, this concept is strange because it is not as commonplace as it once was (like a slide rule).
 * 1) When the point is on the circle:
 * 2) 	Draw ray from center through point
 * 3) 	Make circle centered at point through center
 * 4) 	Draw and circle centered at center
 * 5) 	Same size circle centered at other intersection of circle with ray.
 * 6) 	Connect intersections of circles as tangent
 * 7) When the point is off the circle:
 * 8) 	Draw segment from circle to point
 * 9) 	Draw circles at center and point with segment as radius
 * 10) 	Connect to make perpendicular bisector
 * 11) 	Draw circle with center as midpoint of segment to center of circle

Real-life Applications

 * 1) Euclid's Elements used constructions and were effectively the only mathematical textbook for 2500 years.
 * 2) Problems involving constructions sometimes required very sophisticated methods of solution, for example, squaring a circle was proved to be impossible using Galois Theory within the last century.