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Exploring Circle Properties: Tangents, Intersections, and Relationships in Geometry

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In today's warm-up session, we delve into the unique properties of circles, focusing on tangents, angles, and intersections. Students will review the following concepts: tangent lines that intersect circles, relationships between angles, and how circles can intersect at various points. Key problems will include finding unknown lengths using the Pythagorean theorem and understanding the significance of concentric circles. Students are also encouraged to explore practical applications and problem-solving strategies related to circles in geometric contexts.

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Exploring Circle Properties: Tangents, Intersections, and Relationships in Geometry

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  1. Warm – up Session 48

  2. EOCT Review Question of the Day

  3. Homework Review

  4. CCGPS GeometryDay 48 (3-28-14) UNIT QUESTION: What special properties are found with the parts of a circle? Standard: MMC9-12.G.C.1-5,G.GMD.1-3 Today’s Question: What other angle relationships do we know for circles? Standard: MMC9-12.G.C.2

  5. TANGENT: a LINE that intersects the circle exactly ONE time

  6. Point of Tangency

  7. Two circles can intersect… • in two points • one point • or no points

  8. No points of intersection (different center)

  9. No points of intersection (same center) Concentric Circles Same center but different radii

  10. 1 point of intersection(Tangent Circles) Externally Tangent Internally Tangent

  11. 2 points of intersection

  12. More Pythagorean Theorem type problems! Yeah!!  Point of Tangency If a line (segment or ray) is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

  13. 1. Find x A 12 B 9 a2 + b2 = c2 x 92 + 122 = x2 x = 15

  14. 2. Find RQ a2 + b2 = c2 122 + (QR)2 = (8+12)2 P 122 + (QR)2 = 202 12 8 RQ = 16 R Q

  15. 3. Is BC tangent to the circle? a2 + b2 = c2 ? No A 32 C 16 24 B

  16. r2 + 242 = (r + 16)2 4. Find the radius. r2 + 576= r2 + 32r + 256 320= 32r 16 r = 10 A C 24 B

  17. A green on a golf course is in the shape of a circle. Your golf ball is 8 feet from the edge of the green and 32 feet from a point of tangency on the green. • What is the radius? • b) How far is your ball from the cup at the center? x = 60 ft. x = 68 ft.

  18. S Party hat problems! If two segments from the same exterior point are tangent to a circle, then they are congruent. R T

  19. 6. Find x R S T

  20. 7. Find x R S T

  21. 8. Find x C A B

  22. 9. Find x. B A C P D E

  23. 10. Find NP N T S P R Q

  24. CW Workbook Page 199 #18-33

  25. HW Worksheet

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