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The Power Theorems 10.8
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.
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.
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
BC
D
F
Sample Problem #1
Direction Manual:
1. Label all chords.
2. CRITICAL! STEP!Notice that
A D
C B
F
10
58
x
DFCFBFAF
HOW TO SOLVE THIS EASY PROBLEM: From Math Made Easy
Continued
Therefore, if
Then,10 x 8 is =5x 80=5x X=16
DFCFBFAF
SEE As Easy AS Pie!
10 x
58
A
CB
D
F
Theorem 96
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
P
AT
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.
HExternal part
SAMPLE PROBLEMS ! !!!!M
A
T
H
5
How would you solve this problem?
15
x
Find HT
2
How to Solve this similarly easy problem From Making Hard Math Really Really Fun!
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
AT
HX
2
5
15
THE LAST POWER THEOREM 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.
WHAT IT MEANS
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.
SA
D
R
Q
SAMPLE PROBLEM
RO M
E
O
7 23
x
Find the value of x.
HOW TO SOLVE PROBLEMS with Power Theorems from the author of the book What are these numbers? and What are they doing there ?.
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.
RMO
E
O
723
x
A
B
D
12 cm14 cm
•Find AB =
Answer:
Square root of 312
PRACTICE PROBLEMSPractice Problems
C 4 4P
A3
x
What is the length of PB ?
The length of PB is 1.
PRACTICE PROBLEM # 2More Practice Problems
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?id=L700.
Bibliography
Illuminations: the power of points.2008. NationalCouncil of teachers of mathematics. 30 may 2008.<http://illuminations.nctm.org/LessonDetail.aspx?id=L700>.
Mr.Brown’s Honors Geometry web page.30 may 2008.
<http://www.rockfordschools.org/staff/brownk/Geometryhtm>.
Fogiel. The High School Geometry Tutor. Piscataway, New Jersey.
Research and Education Association.2000.
Bibliography continued
Rhoad,Richard, George Milauslas, Robert Whipple. Geometry for
Enjoyment and Challenge. Evanston,Illinois: McDougal, Littlel
& Company. 2004.
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