Chapter 11 Circles.

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Presentation transcript:

Chapter 11 Circles

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: Point of Tangency: 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 line is perpendicular to the radius drawn to the point of tangency.

Example 1 Finding Angle Measures A) 𝑀𝐿 and 𝑀𝑁 are tangent to ⨀ O. Find the value of x.

Example 1 Finding Angle Measures B) 𝐸𝐷 is tangent to ⨀ O. Find the value of x.

Example 1 Finding Angle Measures C) 𝐵𝐴 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 𝑀𝐿 tangent to ⨀ N at L? Explain.

Example 3 Finding a Tangent B) If NL = 4, LM = 7 and NM = 8, is 𝑀𝐿 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 𝑃𝐴 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 – 3 The 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 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