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Tangents of Circles

This lab is to explore some properties of tangents to a circles.

Investigation:
In Sketchpad, make the following construction.

• Construct circle A with radius AB.
• Construct point C on the circle and construct secant BC.
• Measure angle BAC.

Move point C around the circle. What happens when the line is a tangent? What conjecture can you make about a tangent and a radius to the point of tangency? Use this conjecture to construct a tangent line.

Investigation:
In Sketchpad, make the following construction.

• Construct circle A with radius AB.
• Construct a line perpendicular to AB through B. Move point B to make sure this line is a tangent.
• Construct another radius AC and a tangent through C.
• Construct the point of intersection and label it D.
• Hide the lines and construct the segments BD and CD. These segments are called tangent segments.
• Measure the tangent segments.

What do you notice about these segments? What congecture can you make about tangent segments from the same exterior point?

A common tangent is one tangent line that is tangent to two or more circles. Common tangents are either externally tangent (the tangent is outside the segment from center to center) or internally tangent (the tangent intersects the segment from center to center).

Exercises:
On a sheet of paper, draw two circles with only the common tangents listed.

1. Two internal and two external tangents.
2. Two external tangents and only one internal tangent.
3. Two external and no internal tangents.
4. Only one external tangent and no internal tangents.
5. No common tangents at all.