CHAPTER 11CIRCLES

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 circle that intersects the circle in exactly one point.

POINT OF TANGENCY:The point where a circle and a tangent intersect

A Tangent

Point of Tangency

THEOREM 11 – 1If 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

A) and are tangent to O. Find the value of x.

EXAMPLE 1 FINDING ANGLE MEASURES

B) is tangent to O. Find the value of x.

EXAMPLE 1 FINDING ANGLE MEASURES

C) is tangent to C at point A. Find the value of x.

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 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 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 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

A) Is tangent to N at L? Explain.

EXAMPLE 3 FINDING A TANGENT

B) If NL = 4, LM = 7 and NM = 8, is tangent to N at L? Explain.

EXAMPLE 3 FINDING A TANGENT

C) O has radius 5. Point P is outside O such that PO = 12, and Point A is on O such that PA = 13. Is tangent to O at A? Explain.

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

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

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

EXAMPLE 5 CIRCLES INSCRIBED IN POLYGONS

A) O is inscribed in ∆ ABC. Find the perimeter of ∆ABC.

EXAMPLE 5 CIRCLES INSCRIBED IN POLYGONS

B) O is inscribed in ∆ PQR. ∆PQR has a perimeter of 88 cm. Find QY.

EXAMPLE 5 CIRCLES INSCRIBED IN POLYGONS

C) C is inscribed in quadrilateral XYZW. Find the perimeter of XYZW.

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

HOMEWORK11 – 1 Ditto; # 1 – 15 Odds