1. The Power Theorems 10.8

2.  If two chords of a circle intersect inside the circle, then the product of the measures of the segments of one chord is equal to the product of the measures of the segments of the other chord.  This is also known as the Chord-Chord power theorem.

3.  Before you get your head tied in a knot, let’s examine what this means.  This means that in a circle, if I draw two chords. (A chord is a segment that ends at the endpoints of a circle). Then I cut both in two -The product one set will equal the product of the other set.  Let me explain further with a sample problem.

4.  Find the measure of segment FD if AF= 10 and CF is =5,DF is 8 and BF is simply labeled with the variable x. A B C D F Sample Problem #1

5. Direction Manual: 1.Label all chords. 2. CRITICAL! STEP!Notice that A D C B F 10 5 8 x HOW TO SOLVE THIS EASY PROBLEM: From Math Made Easy

6.  Therefore, if  Then,10 x 8 is =5x  80=5x  X=16 SEE As Easy AS Pie! 10 x 5 8 A C B D F

7.  If a tangent segment and a secant segment are drawn from an external point to a circle, then the square of the measure of the tangent segment is equal to the measures of the entire secant segment and it’s external part.  A.K.A the secant Power Theorem

8. P A T Explaining The Theorem In this circle, PA is a secant segment. TA is a tangent segment. This theorem means that tangent TA squared is equal to AR x HP. The tangent segment squared is equal to the entire secant multiplied by it’s external part. H External part

9. M A T H 5 How would you solve this problem? 15 x Find HT 2

10.  The answer is 10.  MT x AT= 100.  You add 15+5 which will = 20. Then you multiply 20 by 5.  Because of this theorem you take the square root of 100.  The square root of 100 is 10. M A T H X 2 5 15

11.  This is so sad. This is the final theorem. It’s almost over. (Sniff. Sniff).  Oh well.  This theorem is called the secant-secant theorem.  If two secant segments are drawn from an external point on the circle, then the product of the measures of the other secant segment and it’s external part is equal to the product of the measures of the other secant and it’s external part.

12.  The secant secant theorem means  In the yellow circle secant SA multiplied by it’s external part (RA) is equal to DA multiplied by QA. S A D R Q

13. R O M E O 7 2 3 x Find the value of x.

14.  Step 1  Find the total measure of both secants. 7+2=9. RM is 9.OM is equal to 3+x.  Step 2  Find the measure of one secant. RM is equal to  9x2. 9x2=18.  Step 3  Find the measure of the other segment. Since the measure of RM and it’s external part is 18. The Measure of OM is 18/ 3x. The answer to that is 6. The value of x is 6. R M O E O 7 2 3 x

15. A B D 12 cm 14 cm Find AB = Answer: Square root of 312 PRACTICE PROBLEMS Practice Problems

16. C 4 4 P A 3 x What is the length of PB ? The length of PB is 1. PRACTICE PROBLEM # 2 More Practice Problems

17. More practice problems  More practice problems can be found on pages 493-498 in the geometry book.  A useful site for practicing more of these problems is http://illuminations.nctm.org/LessonDetail.aspx?i d=L700http://illuminations.nctm.org/LessonDetail.aspx?i d=L700.

18. Illuminations: the power of points.2008. National Council of teachers of mathematics. 30 may 2008.. Mr.Brown’s Honors Geometry web page.30 may 2008.. Fogiel. The High School Geometry Tutor. Piscataway, New Jersey. Research and Education Association.2000.

19. Rhoad,Richard, George Milauslas, Robert Whipple. Geometry for Enjoyment and Challenge. Evanston,Illinois: McDougal, Littlel & Company. 2004.