Geometry 9 2 Tangents Tangent Definition A line

  • Slides: 16
Download presentation
Geometry 9. 2 Tangents

Geometry 9. 2 Tangents

Tangent Definition A line in the plane of a circle that intersects the circle

Tangent Definition A line in the plane of a circle that intersects the circle in exactly one point. ta ng en t

Theorem If a line is tangent to a circle, then the line is perpendicular

Theorem If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency. ●

Converse of the previous Theorem If a line in the plane of a circle

Converse of the previous Theorem If a line in the plane of a circle is perpendicular to a radius at its outer endpoint, then the line is tangent to the circle. ●

Corollary Tangents to a circle from a point are congruent. ● ●

Corollary Tangents to a circle from a point are congruent. ● ●

Exercises A 1. If OC = 15 and OB = 9, then BC =

Exercises A 1. If OC = 15 and OB = 9, then BC = 12 D 3 -4 -5 Triple 15 O 9 C B 2. If OC = 3√ 6 and BC = 6, then OB = 3√ 2 3√ 6 x²+ 6² = (3√ 6)² x²+ 36 = 54 x² = 18 x = √ 18 = 3√ 2

Exercises 6. If m<COB = 60 and CB = 6√ 3, then AB =

Exercises 6. If m<COB = 60 and CB = 6√ 3, then AB = 12 60˚ 6√ 3 A D 30 -60 -90 Rt. ∆ O C 7. If m<BCD = 70, then m<CBD = m<____ CDB = 55 70˚ Iso. ∆ B

Answers to Exercises A 3. 4. 5. 8. 10 6√ 2 8 25 D

Answers to Exercises A 3. 4. 5. 8. 10 6√ 2 8 25 D O C B

Common Tangents A line that is tangent to each of two coplanar circles. common

Common Tangents A line that is tangent to each of two coplanar circles. common tangent ● ● co mm on ta n ge nt

Internal vs. External Tangents A common external tangent does not. common external tangent ●

Internal vs. External Tangents A common external tangent does not. common external tangent ● ● co A common internal tangent intersects the segment joining the centers of the circle. mm on int er na lt an ge nt

Tangent Circles Two coplanar circles that are tangent to the same line at the

Tangent Circles Two coplanar circles that are tangent to the same line at the same point. There are two types: internally tangent circles externally tangent circles

Circumscribed Polygons A polygon is circumscribed about a circle if each of its sides

Circumscribed Polygons A polygon is circumscribed about a circle if each of its sides is tangent to the circle. A circle is inscribed in a polygon if each of the sides of the polygon is tangent to the circle.

Exercises a. Tell whether the circles are externally tangent, internally tangent, or not tangent.

Exercises a. Tell whether the circles are externally tangent, internally tangent, or not tangent. b. Tell whether the line is a common external tangent, a common internal tangent, or neither. 9. a. not tangent b. com. ext. tangent 10. a. int. tangent b. com. ext. tangent 11. a. not tangent b. neither

Exercises a. Tell whether the circles are externally tangent, internally tangent, or not tangent.

Exercises a. Tell whether the circles are externally tangent, internally tangent, or not tangent. b. Tell whether the line is a common external tangent, a common internal tangent, or neither. 12. a. ext tangent b. com. int. tangent 13. a. not tangent b. com. int. tangent 14. a. not tangent b. com ext. tangent

Exercises a. Tell whether the circles are externally tangent, internally tangent, or not tangent.

Exercises a. Tell whether the circles are externally tangent, internally tangent, or not tangent. b. Tell whether the line is a common external tangent, a common internal tangent, or neither. 15. Find AC 3 A 1 2 B C 24 S 23 R 24 P 13 23 15 AC = 8 17. Find PT and QS 16. Find SP and RQ. P 15 17 13 Q 17 Q SP = 39 RQ = 40 PT = 22 QS = 22 9 S R 9 T

Homework pg. 335 WE #1 -6, 8 -10, 14 -18

Homework pg. 335 WE #1 -6, 8 -10, 14 -18

Tangent Line Recall from geometry Tangent is aTangent Line Recall from geometry Tangent is a
Tangent Lines Understanding Tangent Lines A tangent toTangent Lines Understanding Tangent Lines A tangent to
Tangent Line Approximations A tangent line approximation isTangent Line Approximations A tangent line approximation is
Section 10 5 Tangents Geometry Definitions Tangent aSection 10 5 Tangents Geometry Definitions Tangent a
2 72 8 TANGENT LINES DERIVATIVES TANGENT LINE2 72 8 TANGENT LINES DERIVATIVES TANGENT LINE
Tangent lines n Recall tangent line is theTangent lines n Recall tangent line is the
The Slope of the Tangent Line The TangentThe Slope of the Tangent Line The Tangent
Tangents Use properties of tangents Solve problems involvingTangents Use properties of tangents Solve problems involving
Inscribed Angles Tangents Angles Secants Tangents Angles SegmentsInscribed Angles Tangents Angles Secants Tangents Angles Segments
Tangents To recognize tangents and use the propertiesTangents To recognize tangents and use the properties
9 5 Tangents Objectives To recognize tangents and9 5 Tangents Objectives To recognize tangents and
36 Tangents to Circles Tangents to Circles Std36 Tangents to Circles Tangents to Circles Std
Projective Geometry Projective Geometry Projective Geometry Projective GeometryProjective Geometry Projective Geometry Projective Geometry Projective Geometry
Tangents An external tangent of two circles DrawTangents An external tangent of two circles Draw
Tangents An interior tangent of two circles DrawTangents An interior tangent of two circles Draw