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Circle G E O M E T R Y

Circle G E O M E T R Y. Radius (or Radii for plural). The segment joining the center of a circle to a point on the circle. Example: OA. Diameter. A chord that passes through the center of a circle. Example: AB. Chord. A segment joining two points on a circle Example: AB. Chord.

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Circle G E O M E T R Y

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  1. Circle GEOMETRY

  2. Radius (or Radii for plural) • The segment joining the center of a circle to a point on the circle. • Example: OA

  3. Diameter • A chord that passes through the center of a circle. • Example: AB

  4. Chord • A segment joining two points on a circle • Example: AB

  5. Chord • A segment joining two points on a circle • Example: AB

  6. Secant • A line that intersects the circle at exactly two points. • Example: AB

  7. Tangent • A line that intersects a circle at exactly one point. • Example: AB

  8. Arc • A figure consisting of two points on a circle and all the points on the circle needed to connect them by a single path. • Example: arc AB

  9. Diagram of Arcs

  10. Central Angle • An angle whose vertex is at the center of a circle. • Example: Angle ABC

  11. Central angles will always equal the inscribed arc. • Example: angle ABC = arc AC

  12. Example 2 • Find the measures of the red arcs. Are the arcs congruent?

  13. Example 1 • Find the measure of each arc. 70° 360° - 70° = 290° 180°

  14. Inscribed Angle • An angle whose vertex is on a circle and whose sides are determined by two chords. • Example: Angle ABC

  15. Intercepted Arc • An arc that lies in the interior of an inscribed angle. • Example: arc AC

  16. An inscribed arc will always equal twice the inscribed angle. • Ex. Arc AC= 2 times Angle ABC

  17. Example 1 • Find the measure of the blue arc or angle. a. b.

  18. Example 1 • Tell whether the line or segment is best described as a chord, a secant, a tangent, a diameter, or a radius. tangent diameter chord radius

  19. Tangent Theorem • The tangent is a line or line segment that touches the perimeter of a circle at one point only and is perpendicular to the radius that contains the point. 

  20. Example 3 Use the converse of the Pythagorean Theorem to see if the triangle is right. 112 + 432 ? 452 121 + 1849 ? 2025 1970  2025

  21. Definitions • Inscribed polygon – a polygon whose vertices all lie on a circle. • Circumscribed circle – A circle with an inscribed polygon. The polygon is an inscribed polygon and the circle is a circumscribed circle.

  22. Inscribed Quadrilateral • If a quadrilateral is inscribed in a circle, then both pairs of opposite angles are supplementary.

  23. 1. Problem: Find the measure of arc GDE.

  24. Inscribed Right Triangle Theorem • If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle.

  25. Example 3 • Find the value of each variable. b. a.

  26. Chord Product Theorem • If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.

  27. Example 1 • Find the value of x.

  28. Try This! • Find the value of x.

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