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Understanding Circles in Geometry

Learn key terms like radius, chord, diameter, secant, tangent, and theorems about tangents to enhance your grasp of circle properties. Explore concepts such as concentric and tangent circles.

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Understanding Circles in Geometry

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  1. Section 9-1 Circles Vocab

  2. A circle is named by its center point. For example: Circle A or A. Circle – the set of all points in a plane a given distance away from a center point. A Radius (r) Plural: Radii Radius – the “given distance away from the center point” of a circle; a segment that joins the center to a point on the circle.

  3. Sphere – the set of all points a given distance away from a center point.

  4. Chord – a segment whose endpoints lie on on the circle. Example: DC A B Diameter – a chord that passes through the center of the circle. Example: AB A diameter is twice the length of a radius. C D

  5. Example: AB Secant – a line that contains a chord. B A **Note: A chord and a secant can be named using the same letters. The notation tells you whether it is a secant or a chord. A secant is a line; a chord is a segment.** Secant: AB Chord: AB

  6. Example: AB Tangent – a line that intersects a circle at exactly one point. Not a tangent! B The point at which the circle and the tangent intersect is called the point of tangency. Example: A A

  7. Congruent Circles – circles with congruent radii. 5cm 5cm Concentric Circles – circles with the same center point.

  8. When each vertex of a polygon is on the circle, the circle is said to be circumscribed around the polygon. This circle is circumscribed around the pentagon

  9. When each side of a polygon is tangent to a circle, the circle is said to be inscribed in the polygon. This circle is inscribed inside of the pentagon.

  10. Section 9-2 Tangents

  11. If AB is tangent to Circle Q at point C, then QC ^ AB. 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. Q is the center of the circle. C is a point of tangency. Q A C B

  12. Example: Given Circle Q with a radius length of 7. D is a point of tangency. DF = 24, find the length of QF. Q 7 D 24 F 72 + 242 = QF2 QF = 25 G NOTE: G is NOT necessarily the midpoint of QF!! Extension: Find GF. GF = 18 QF = 25 QG = 7

  13. Theorem 9-2: 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. This is the converse of Theorem 9-1.

  14. Tangent Circles – coplanar circles that are tangent to the same line at the same point. Internally Tangent Circles Externally Tangent Circles

  15. Common Tangent – a line that is tangent to two coplanar circles. Common Internal Tangent Intersects the segment joining the centers.

  16. Common External Tangent Does not intersect the segment joining the centers.

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