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Warm-Up

Warm-Up. Given circle O has a radius of 12 in., and PR is a diameter:. Q. 1. Find mQRS. R. P. O. 2. Find mQPS. S. 3. Find the circumference of the circle. 4. Find the length of RS.

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Warm-Up

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  1. Warm-Up Given circle O has a radius of 12 in., and PR is a diameter: Q 1. Find mQRS. R P O 2. Find mQPS. S 3. Find the circumference of the circle. 4. Find the length of RS. 5. Name 1 minor arc, 1 major arc, and one semi-circle.

  2. 12.2 Properties of Tangents Geometry

  3. Objectives/Assignment • Identify segments and lines related to circles. • Use properties of a tangent to a circle.

  4. The distance across the circle, through its center is the diameter of the circle. The diameter is twice the radius. The terms radius and diameter describe segments as well as measures. Some definitions you need

  5. A radius is a segment whose endpoints are the center of the circle and a point on the circle. QP, QR, and QS are radii of Q. All radii of a circle are congruent. Some definitions you need

  6. A chord is a segment whose endpoints are points on the circle. PS and PR are chords. A diameter is a chord that passes through the center of the circle. PR is a diameter. Some definitions you need

  7. A secant is a line that intersects a circle in two points. Linek is a secant. A tangent is a line in the plane of a circle that intersects the circle in exactly one point. Linej is a tangent. Some definitions you need

  8. Identifying Special Segments and Lines • Tell whether the line or segment is best described as a chord, a secant, a tangent, a diameter, or a radius of C. • AD • CD • EG • HB

  9. In a plane, two circles can intersect in two points, one point, or no points. Coplanar circles that intersect in one point are called tangent circles. Coplanar circles that have a common center are called concentric. More information you need-- 2 points of intersection.

  10. A line or segment that is tangent to two coplanar circles is called a common tangent. A common internal tangent intersects the segment that joins the centers of the two circles. A common external tangent does not intersect the segment that joins the center of the two circles. Tangent circles Internally tangent Externally tangent

  11. Circles that have a common center are called concentric circles. Concentric circles No points of intersection Concentric circles

  12. Using properties of tangents • The point at which a tangent line intersects the circle to which it is tangent is called the point of tangency. You will justify theorems in the exercises.

  13. If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. If l is tangent to Q at point P, then l ⊥QP. Theorem 12.2 l

  14. If l ⊥QP at P, then l is tangent to Q. Converse of Theorem 12.2 l

  15. You can use the Converse of the Pythagorean Theorem to tell whether EF is tangent to D. Because 112 + 602 = 612, ∆DEF is a right triangle and DE is perpendicular to EF. So by Theorem 10.2; EF is tangent to D. Verifying a Tangent to a Circle

  16. You are standing at C, 8 feet away from a grain silo. The distance from you to a point of tangency is 16 feet. What is the radius of the silo? First draw it. Tangent BC is perpendicular to radius AB at B, so ∆ABC is a right triangle; so you can use the Pythagorean theorem to solve. Ex. 5: Finding the radius of a circle

  17. Solution: c2 = a2 + b2 Pythagorean Thm. (r + 8)2 = r2 + 162 Substitute values r 2 + 16r + 64 = r2 + 256 Square of binomial Subtract r2 from each side. 16r + 64 = 256 Subtract 64 from each side. 16r = 192 r = 12 Divide. The radius of the silo is 12 feet.

  18. Note: • From a point in the circle’s exterior, you can draw exactly two different tangents to the circle. The following theorem tells you that the segments joining the external point to the two points of tangency are congruent.

  19. If two segments from the same exterior point are tangent to the circle, then they are congruent. IF SR and ST are tangent to P, then SR  ST. Theorem 10.3

  20. Proof of Theorem 10.3 • Given: SR is tangent to P at R. • Given: ST is tangent to P at T. • Prove: SR  ST

  21. Statements: SR and ST are tangent to P SR  RP, STTP RP = TP RP  TP PS  PS ∆PRS  ∆PTS SR  ST Reasons: Given Tangent and radius are . Definition of a circle Definition of congruence. Reflexive property HL Congruence Theorem CPCTC Proof

  22. AB is tangent to C at B. AD is tangent to C at D. Find the value of x. Ex. 7: Using properties of tangents x2 + 2

  23. x2 + 2 Solution: AB = AD Two tangent segments from the same point are  11 = x2 + 2 Substitute values 9 = x2 Subtract 2 from each side. 3 = x Find the square root of 9. The value of x is 3 or -3.

  24. Homework • Complete Worksheet

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