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Chapter 10 CIRCLES

Chapter 10 CIRCLES. Ms. Watson Geometry Banneker Academic High School. 10.1 Tangents to Circles. GOAL #1 Identify segments and lines related to circles. Who’s N’ The Circle Fam ?. radius. diameter. Who’s N’ The Circle Fam ?. secant. R. Q. tangent. P. S.

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Chapter 10 CIRCLES

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  1. Chapter 10CIRCLES Ms. Watson Geometry Banneker Academic High School

  2. 10.1 Tangents to Circles

  3. GOAL #1Identify segments and lines related to circles.

  4. Who’s N’ The Circle Fam? radius diameter

  5. Who’s N’ The Circle Fam? secant R Q tangent P S

  6. Who’s N’ The Circle Fam? Common External Tangent Common Internal Tangent Common Tangents A line or line segment that is tangent to two circles in the same plane is called a common tangent. 2 Types

  7. GOAL #2Use properties of tangent to a circle.

  8. Theorem 10.1 If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. What do we know about right triangles??? How can we use what we know to solve the length of sides of a triangle???

  9. Example 1

  10. Answer

  11. Example 2

  12. Answer

  13. Theorem 10.3 If two segments from the same exterior point are tangent to a circle, then they are congruent (equal).

  14. 10.2 Arcs & Chords

  15. GOAL #1Use properties of arcs of circles.

  16. Who’s N’ The Circle Fam? A Minor Arc C K Major Arc B

  17. Name That Arc!Arcs are named by their endpoints. A Minor Arc C K Major Arc B

  18. Measure That Arc! 60 ° 360° - 60 ° = 300 ° A Minor Arc 60 ° C 60 ° 300 ° K Major Arc B

  19. Arc Addition PostulateDiscovery Step 1: Draw a circle. Step 2: Place three points on the circle named A, B, and C. Discover that… The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. mABC = mAB + mBC

  20. Arc for Thought • Two arcs of the same circle or of congruent circles are congruent arcs if they have the same measure. So, two minor arcs of the same circle or of congruent circles are congruent if their central angles are congruent. Which minor arcs are congruent? Why are they congruent?

  21. GOAL #2Use properties of chords of circles.

  22. Theorem 10.4 A C B

  23. Theorem 10.5

  24. Theorem 10.6 If one chord is a perpendicular bisector of another chord, the first chord is a diameter. J M K L

  25. Theorem 10.7

  26. Hands on Activity

  27. 10.3 Inscribed Angles

  28. GOAL #1Use inscribed angles to solve problems.

  29. Who’s N’ The Circle Fam?

  30. Theorem 10.8 Measure of an Inscribed Angle If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc. m ABC = mAC A C B

  31. Theorem 10.9 If two inscribed angles of a circle intercept the same are, then the angles are congruent. A C D B C D

  32. GOAL #2Use properties of inscribed polugons.

  33. Who’s N’ The Circle Fam? If all of the vertices of a polygon lie on a circle, the polygon is INSCRIBED in the circle and the circle is CIRCUMSCRIBED about the polygon.

  34. Theorem 10.10 If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. Also the angle opposite the diameter is a right angle. B is a right angle if and only if is a diameter of the circle. A B C

  35. Theorem 10.11 A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary. D, E, F, and G lie on the circle if and only if E F m D + m F = 180 And m E + m G = 180 G D

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