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Investigating Properties of Tangents to a Circle

Consider a circle with center and point outside of the circle. Using a straightedge and compass, can you construct a tangent to the circle through the given point?

Circle and External Point
Think about the properties of tangents. How can you justify that the line you draw is indeed the tangent?


Drawing a Tangent Line to a Circle Through an Outer Point


External Tangent Congruence Theorem


External Tangent Congruence Theorem

In the diagram, all three segments are tangent to circle

The points and are the points where the segments touch the circle. If and find


From the graph, it can be seen that and are tangent segments with a common endpoint outside By the External Tangent Congruence Theorem, and are congruent.

Smilarly, and and and are congruent tangent segments.
Using the Segment Addition Postulate and the given lengths, a system of equations with three equations and three unknowns can be written.
To find the Elimination Method can be used. Start by multiplying the second equation by
Adding this equation to the first equation will eliminate
Since and are additive inverses, adding this equation to the third equation will eliminate and thus give


Tangents to Circles in Real Life

Imagine a superhero joining the Olympics to throw a hammer. An athlete would typically spin counterclockwise three or four (rarely five) times, then release the hammer. As viewed from above, the hammer travels on a path that is tangent to the circle created when the athlete spins. The diagram below shows the path of the superhero's hammer throw. See how the super hero fairs! Note, it is a not-to-scale drawing.
On August in a packed stadium full of fans, Yuriy Sedykh set the world record with a throw of meters. Perhaps a throw farther than the superhero!