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10.5 Tangents & Secants

10.5 Tangents & Secants. Objectives. Use properties of tangents Solve problems using circumscribed polygons. k. j. Tangents and Secants. A tangent is a line in the plane of a circle that intersects the circle in exactly one point. Line j is a tangent.

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10.5 Tangents & Secants

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  1. 10.5 Tangents & Secants

  2. Objectives • Use properties of tangents • Solve problems using circumscribed polygons

  3. k j Tangents and Secants • A tangent is a line in the plane of a circle that intersects the circle in exactly one point. Line j is a tangent. • A secant is a line that intersects a circle in two points. Line k is a secant. A secant contains a chord.

  4. r j Tangents Theorem 10.9: If a line is tangent to a , then it is ┴ to the radius drawn to the point of tangency. The converse is also true. r ┴ j

  5. ALGEBRA is tangent to at point R. Find y. Because the radius is perpendicular to the tangent at the point of tangency, . This makes a right angle and  a right triangle. Use the Pythagorean Theorem to find QR, which is one-half the length y. Example 1:

  6. Answer: Thus, y is twice . Example 1: Pythagorean Theorem Simplify. Subtract 256 from each side. Take the square root of each side. Because y is the length of the diameter, ignore the negative result.

  7. is a tangent to at point D. Find a. Your Turn: Answer: 15

  8. Determine whether is tangent to Example 2a: First determine whether ABCis a right triangle by using the converse of the Pythagorean Theorem.

  9. Answer: So, is not tangent to . Example 2a: Pythagorean Theorem Simplify. Because the converse of the Pythagorean Theorem did not prove true in this case, ABC is not a right triangle.

  10. Determine whether is tangent to Example 2b: First determine whether EWDis a right triangle by using the converse of the Pythagorean Theorem.

  11. Answer: Thus, making a tangent to Example 2b: Pythagorean Theorem Simplify. Because the converse of the Pythagorean Theorem is true, EWD is a right triangle and EWD is a right angle.

  12. a. Determine whether is tangent to Your Turn: Answer: yes

  13. b. Determine whether is tangent to Your Turn: Answer: no

  14. W Z X Y More about Tangents Theorem 10.11:If two segments from the same exterior point are tangent to a circle, then they are congruent. XW  XY

  15. are drawn from the same exterior point and are tangent to so are drawn from the same exterior point and are tangent to Example 3: ALGEBRA Find x. Assume that segments that appear tangent to circles are tangent.

  16. Example 3: Definition of congruent segments Substitution. Use the value of y to find x. Definition of congruent segments Substitution Simplify. Subtract 14 from each side. Answer: 1

  17. Your Turn: ALGEBRA Find a. Assume that segments that appear tangent to circles are tangent. Answer: –6

  18. Triangle HJK is circumscribed about Find the perimeter of HJK if Example 4:

  19. We are given that Example 4: Use Theorem 10.10 to determine the equal measures. Definition of perimeter Substitution Answer: The perimeter of HJKis 158 units.

  20. Triangle NOT is circumscribed about Find the perimeter of NOT if Your Turn: Answer: 172 units

  21. Assignment • Pre-AP GeometryPg. 556 #8 – 20, 23 - 26 • Geometry:Pg. 556 #8 – 18, 23 - 25

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