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Secants and Tangents Section 10.4

Secants and Tangents Section 10.4. By: Matt Lewis. Secants and Tangents. -Objectives Identify secant and tangent lines and segments. Distinguish between two types of tangent circles. Recognize common internal and common external tangents. Definitions.

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Secants and Tangents Section 10.4

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  1. Secants and TangentsSection 10.4 By: Matt Lewis

  2. Secants and Tangents -Objectives • Identify secant and tangent lines and segments. • Distinguish between two types of tangent circles. • Recognize common internal and common external tangents.

  3. Definitions • Secant- a line that intersects a circle a exactly two points. Every secant contains a chord of the circle. • Tangent- a line that intersects a circle at exactly one point. This point of contact is called the point of tangency. P . M . Secant . J Tangent

  4. Definitions Con’t • Tangent Segment- Part of a tangent line between the point of contact and a point outside the circle. • Secant Segment- Part a secant line that joins a point outside the circle to the farther intersection point of the secant and the circle. • External Part of a secant segment- the part of a secant line that joins the outside point to the nearer intersection point. Tangent Segment . . L M . T . A . Secant Segment External Part M .

  5. Definitions Con’t • Tangent Circles- circles that intersect each other at exactly one point. • Externally Tangent Circles- each of the tangent circles lies outside the other. • Internally Tangent Circles- one of the tangent circles lies inside the other. M . A T - - - - - - - - - - - - - - - - - - M A T

  6. Definitions Con’t • Common Tangent- a line tangent to two circles. • Common Internal Tangent- the tangent lies between the circles. ( WI ) • Common External Tangent- the tangent is not between the circles. ( LM ) I S E W M L

  7. Postulates & Theorems • Postulates • A tangent line is perpendicular to the radius drawn to the point of contact. • If a line is perpendicular to a radius at its outer endpoint then it is tangent to the circle. • Theorems • If two tangent segments are drawn to a circle from an exterior point, then those segments are congruent.

  8. Common Tangent Procedure • Draw the segment joining the centers. • Draw the radii to the points of contact. • Through the center of the smaller circle, draw a line parallel to the common tangent. • Observe that this line will intersect the radius of the larger circle (extended if necessary) to form a rectangle and a right triangle. • Use the Pythagorean Theorem and properties of a rectangle.

  9. Sample Problems Sample Problem #1 Step 1 - Constructing radius PB at the point of tangency as shown. Since lengths of all the radii of a circle are equal, PB = 8. Step 2 - Since the tangent and the radius at the point of tangency are always perpendicular, ΔABP is a right angled triangle. Step 3 - Using the Pythagorean theorem, Step 4 - Substituting for AP, AB and BP, Step 5 - Since the negative value of the square root will yield a negative value for x, taking the positive square root of both sides, x = 9. Given: AC is Tangent to circle P Calculate the value of X.

  10. Sample Problems Sample Problem #2 Solution: OA is AP and OB PB. A = 90 O = 90 140 AOBP is a quadrilateral. 90 + 90 + 140 + X = 360 X = 40 P B PA and PB are Tangents to Circle O. Find:

  11. Practice Problems #1 #2 Find: a, b, and c. JK is tangent to circles Q & P. Find: JK

  12. Practice Problems #3 Given: Two tangent circles, is a common external tangent, is the common internal tangent. Prove: D is the midpt. of BC.

  13. Practice Problems #4 R P S OS = 20 Q PS = 12 O What is QS?

  14. Answers to Practice Problems #3 • #1- JK = 20. • #2- = 65 = 25 = 65 • #4- QS = 4 Statements Reasons • Two circles are • externally tangent 1. Given 2. BC is a common external tangent. 2. Given 3. DA is a common internal tangent. 3. Given 4. Any two tangents drawn to a circle from the same point are . 4. DB DA 5. DC DA 5. Same as 4. 6. DB DC 6. Transitive 7. If a point divides a line into two seg., then it is the midpt. 7. D is the midpt. of BC.

  15. Practice Exercises • Pg. 463-464 #1,2,5, & 6. • Pg. 464-465 #9,10,11-14,16-18. • These exercises come from our book.

  16. Works Citied Rhoad, Richard. Geometry for Enjoyment and Challenge. Boston: McDougal Littell, 1991. Wolf, Ira. Barron’s SAT Subject Test- Math Level I. Barron Publishing, 2008. Shapes-Circles. http://www.bbc.co.uk/schools/.html. 27 May 2008. Practice Problems Geometry. http://www.hotmath.com, 27 May 2008.

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