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Lesson 9-2 Tangents (page 333)
Essential Question How can relationships in a circle allow you to solve problems involving tangents?
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Theorem 9-1 If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency. O m T
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This theorem is proven with an indirect proof
This theorem is proven with an indirect proof. Also, this is why the tangent term is used in trigonometry. If O is a unit circle, the radius = 1, then … O 1 m T Z
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Tangents to a circle from a point are congruent.
Corollary Tangents to a circle from a point are congruent. A P O B
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The tangents to a circle from a point are really the tangent segments.
B Tangent segments to a circle from a point are congruent.
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To prove this corollary, prove ∆PAO ≅ ∆PBO by the HL Theorem.
Then PA = PB by CPCTC.
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Theorem 9-2 If a line in the plane of a circle is perpendicular to the radius at its outer endpoint, then the line is tangent to the circle. Q l R
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This is the converse of Theorem 9-1 and it also is proven with an indirect proof.
Q l R
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example: RS is tangent to P at R. If PR = 5 and RS = 12, find PS.
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CIRCUMSCRIBED POLYGON
… a polygon is circumscribed about a circle if each of its sides is tangent to the circle.
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INSCRIBED CIRCLE … a circle is inscribed in a polygon
if each of the sides of the polygon is tangent to the circle.
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CIRCUMSCRIBED POLYGON
and INSCRIBED CIRCLE are the same thing!
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COMMON TANGENT … a line that is tangent to each
of two coplanar circles.
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COMMON INTERNAL TANGENT
… intersects the segment joining the centers of the circles.
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Here is another COMMON TANGENT.
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COMMON EXTERNAL TANGENT
… does not intersect the segment joining the centers of the circles.
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Are there anymore COMMON TANGENTS?
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TANGENT CIRCLES … two coplanar circles that are tangent to the
same line at the same point. #1 Externally Tangent Circles
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TANGENT CIRCLES … two coplanar circles that are tangent to the
same line at the same point. #2 Internally Tangent Circles
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Classroom Exercises on page 335
Classroom Assignment Classroom Exercises on page 335 1 to 3 all numbers How can relationships in a circle allow you to solve problems involving tangents?
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Written Exercises on page 335
Assignment Written Exercises on page 335 GRADED: 1 to 5 odd numbers How can relationships in a circle allow you to solve problems involving tangents?
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