CHAPTER 11 CIRCLES SECTION 11 1 TANGENT LINES

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CHAPTER 11 CIRCLES

CHAPTER 11 CIRCLES

SECTION 11 – 1 TANGENT LINES Objectives: • To use the relationship between a

SECTION 11 – 1 TANGENT LINES Objectives: • To use the relationship between a radius and a tangent • To use the relationship between two tangents from one point

A TANGENT TO A CIRCLE: A line, ray or segment in the plane of

A TANGENT TO A CIRCLE: A line, ray or segment in the plane of a circle that intersects the circle in exactly one point. A Tangent Point of Tangency POINT OF TANGENCY: The point where a circle and a tangent intersect

THEOREM 11 – 1 If a line is tangent to a circle, then the

THEOREM 11 – 1 If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency.

EXAMPLE 1 FINDING ANGLE MEASURES

EXAMPLE 1 FINDING ANGLE MEASURES

EXAMPLE 1 FINDING ANGLE MEASURES

EXAMPLE 1 FINDING ANGLE MEASURES

EXAMPLE 1 FINDING ANGLE MEASURES

EXAMPLE 1 FINDING ANGLE MEASURES

EXAMPLE 2 REAL-WORLD CONNECTION A) A dirt bike chain fits tightly around two gears.

EXAMPLE 2 REAL-WORLD CONNECTION A) A dirt bike chain fits tightly around two gears. The chain and gears form a figure like the one at the right. Find the distance between the centers of the gears.

EXAMPLE 2 REAL-WORLD CONNECTION B) A belt fits tightly around two circular pulleys, as

EXAMPLE 2 REAL-WORLD CONNECTION B) A belt fits tightly around two circular pulleys, as shown at the right. Find the distance between the centers of the pulleys.

EXAMPLE 2 REAL-WORLD CONNECTION C) A belt fits tightly around two circular pulleys, as

EXAMPLE 2 REAL-WORLD CONNECTION C) A belt fits tightly around two circular pulleys, as shown at the right. Find the distance between the centers of the pulleys.

THEOREM 11 – 2(CONVERSE OF 11 – 1) If a line in the plane

THEOREM 11 – 2(CONVERSE OF 11 – 1) If a line in the plane of a circle is perpendicular to a radius at its endpoint on the circle, then the line is tangent to the circle.

EXAMPLE 3 FINDING A TANGENT

EXAMPLE 3 FINDING A TANGENT

EXAMPLE 3 FINDING A TANGENT

EXAMPLE 3 FINDING A TANGENT

EXAMPLE 3 FINDING A TANGENT

EXAMPLE 3 FINDING A TANGENT

HOMEWORK: TEXTBOOK PAGE 586; #1 – 4, 6, 10 – 12

HOMEWORK: TEXTBOOK PAGE 586; #1 – 4, 6, 10 – 12

SECTION 11 – 1 CONTINUED… Objectives: • To use the relationship between two tangents

SECTION 11 – 1 CONTINUED… Objectives: • To use the relationship between two tangents from one point

TRIANGLE INSCRIBED IN A CIRCLE TRIANGLE CIRCUMSCRIBED ABOUT A CIRCLE

TRIANGLE INSCRIBED IN A CIRCLE TRIANGLE CIRCUMSCRIBED ABOUT A CIRCLE

THEOREM 11 – 3 The two segments tangent to a circle from a point

THEOREM 11 – 3 The two segments tangent to a circle from a point outside the circle are congruent.

EXAMPLE 5 CIRCLES INSCRIBED IN POLYGONS

EXAMPLE 5 CIRCLES INSCRIBED IN POLYGONS

EXAMPLE 5 CIRCLES INSCRIBED IN POLYGONS

EXAMPLE 5 CIRCLES INSCRIBED IN POLYGONS

EXAMPLE 5 CIRCLES INSCRIBED IN POLYGONS

EXAMPLE 5 CIRCLES INSCRIBED IN POLYGONS

ADDITIONAL EXAMPLES Assume the lines that appear to be tangent are tangent. O is

ADDITIONAL EXAMPLES Assume the lines that appear to be tangent are tangent. O is the center of each circle. Find the value of x.

HOMEWORK 11 – 1 Ditto; # 1 – 15 Odds

HOMEWORK 11 – 1 Ditto; # 1 – 15 Odds

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