Straight Edge and Compass Construction of the Tangent to a Circle Through an External Point

Objective:
Given a circle and an external point, the students will be able to construct a tangent to the circle through the point using a straightedge and compass. They will also be able to explain why this construction works.

Opening Activity:
Have a class discussion about what being tangent means. Also, review that a triangle inscribed in a circle on the diameter of the circle is always a right triangle.

Order of Activities:
Give the students five minutes to work on the construction on their own or in pairs. After five minutes, bring the class together and go through the construction together:

• Construct a circle with center O and any point P outside the circle.
• Use a straightedge to construct the segment OP.
  
  
  
• Find the midpoint of OP by constructing two circles. One circle should have center P and radius OP, and the other circle should have center O and radius OP. Now construct a line l through the intersection points of these two circles. This line will bisect OP. The intersection of line l and segment OP is the midpoint M of OP.
  Click here to see the midpoint construction.
  
  
• Construct a circle with center M and radius OM.
• Find the intersection of this circle and the circle that we started with. Label one of these points Q.
  
  
  
• Construct line PQ. This is the tangent line that we were looking for.
  

Wrap Up and Homework:
Answer the following questions in your math journals: How do you know that PQ is tangent to circle O? Why does this tangent construction work?

Activities for Tomorrow:
Use The Geometer's Sketchpad to consruct the tangent to a circle through an external point.