Tangent Segments

New York State Common Core Math Geometry, Module 5, Lesson 12

Worksheets for Geometry

Student Outcomes

• Students use tangent segments and radii of circles to conjecture and prove geometric statements, especially those that rely on the congruency of tangent segments to a circle from a given point.
• Students recognize and use the fact if a circle is tangent to both rays of an angle, then its center lies on the angle bisector.

Tangent Segments

Classwork

Opening Exercise

In the diagram, what do you think the length of 𝑧 could be? How do you know?

• Show Video

Example

In each diagram, try to draw a circle with center 𝐷 that is tangent to both rays of ∠𝐵𝐴C

In which diagrams did it seem impossible to draw such a circle? Why did it seem impossible? What do you conjecture about circles tangent to both rays of an angle? Why do you think that?

Exercises

1. You conjectured that if a circle is tangent to both rays of a circle, then the center lies on the angle bisector.
   a. Rewrite this conjecture in terms of the notation suggested by the diagram.
   Given:
   Need to show:
   b. Prove your conjecture using a two-column proof.
   
2. An angle is shown below.
   a. Draw at least three different circles that are tangent to both rays of the given angle.
   b. Label the center of one of your circles with 𝑃. How does the distance between 𝑃 and the rays of the angle compare to the radius of the circle? How do you know?
3. Construct as many circles as you can that are tangent to both the given angles at the same time. You can extend the rays as needed. These two angles share a side.
   Explain how many circles you can draw to meet the above conditions and how you know.
4. In a triangle, let 𝑃 be the location where two angle bisectors meet. Must 𝑃 be on the third angle bisector as well?
   Explain your reasoning.
5. Using a straightedge, draw a large triangle 𝐴𝐵𝐶.
   a. Construct a circle inscribed in the given triangle.
   b. Explain why your construction works
   c. Do you know another name for the intersection of the angle bisectors in relation to the triangle?

Lesson Summary

THEOREMS:

• The two tangent segments to a circle from an exterior point are congruent.
• If a circle is tangent to both rays of an angle, then its center lies on the angle bisector.
• Every triangle contains an inscribed circle whose center is the intersection of the triangle’s angle bisectors.

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