2. Tangents and Circles, Part 1 Lesson 58

3. Definitions (Review) A circle is the set of all points in a plane that are equidistant from a given point called the center of the circle. Radius – the segment (distance) from the center to a point on the circle Congruent circles – circles that have the same radius. Diameter – the segment with endpoints on the circle and contains the center

4. Diagram of Important Terms (Review) center

5. Definition (Review) Chord – a segment whose endpoints are points on the circle.

6. Definition (Review) Secant – a line that intersects a circle in two points.

7. Definition (Review) Tangent – a line in the plane of a circle that intersects the circle in exactly one point.

8. Definition (Review) Point of tangency – the point at which a tangent line intersects the circle to which it is tangent point of tangency

9. Review Tell whether the line or segment is best described as a chord, a secant, a tangent, a diameter, or a radius. tangent at Point B diameter chord radius

10. Theorem 58-1 If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency.

11. Example 1 Line r is tangent to ʘ C at point B and line s passes through point C. Lines r and s intersect at point A. a)Sketch and label. b)What is m<CBA? c)If m<BCA = 50°, then what is m<CAB?

12. Example 1a Line r is tangent to ʘ C at point B and line s passes through point C. Lines r and s intersect at point A. a)Sketch and label.

13. Example 1a Line r is tangent to ʘ C at point B and line s passes through point C. Lines r and s intersect at point A. a)Sketch and label.

14. Example 1a Line r is tangent to ʘ C at point B and line s passes through point C. Lines r and s intersect at point A. a)Sketch and label.

15. Example 1b Line r is tangent to ʘ C at point B and line s passes through point C. Lines r and s intersect at point A. b)What is m<CBA? m<CBA = 90 °

16. Example 1c Line r is tangent to ʘ C at point B and line s passes through point C. Lines r and s intersect at point A. c)If m<BCA = 50°, then what is m<CAB? m<CAB + m<BCA = 90 ° m<CAB + 50 ° = 90 ° m<CAB = 40 °

17. Theorem 58-2 If a line in a plane is perpendicular to a radius at its endpoint on the circle, then the line is tangent to the circle.

18. Example 2 Use the converse of the Pythagorean Theorem to see if the triangle is right. 11 2 + 43 2 ? 45 2 121 + 1849 ? 2025 1970  2025

19. Theorem 58-3 If two tangent segments are drawn to a circle from the same exterior point, then they are congruent.

20. Example 3

21. Example 4 a)Find the perimeter of ΔJLM. b)What is the relationship between ΔJLM and ΔJLK? c)What is the perimeter of JKLM?

22. Example 4a a)Find the perimeter of ΔJLM. MJ = 12 in 5 2 + 12 2 = (LJ) 2 25 + 144 = (LJ) 2 169 = (LJ) 2 13 in = LJ P = 12 + 5 +13 P = 30 in

23. Example 4b

24. Example 4c c)What is the perimeter of JKLM? P = 5 + 5 + 12 + 12 P = 34 in

25. Do you have any questions?