geometry
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Geometry
Unit VIII10.5: Segments of Chords
Objective: students will be finding the lengths of segments formed when chords intersect.
In this section, we will be finding the lengths of segments formed when chords intersect. Theorem 10.5: If a diameter is perpendicular to a chord, then it bisects the chord and
its arc.
Example: In OQ, QBAC , AC = 8 and QD = 4. Find the radius of OQ. Example: In a circle of radius 10 inches, a chord is 5 inches from the center. Find the length of the chord as an exact answer.
Q
C
B
A D
Q
C
B
A D44
π2=42+42
π2=32π=β32=4β2
105
102=52+π₯2
π₯2=75
π₯=β75=5β3hπππππ‘ ππ hπ πππ=10 β3
Theorem 10.15: If two chords intersect inside a circle, then the product of the two
segments on one chord equals the product of the two segments on the
second chord.
Example: If AX = 7, XC = 8 and BX = 11, find XD.
Example: If AX = 8, XC = 15, BX = 18 and XD = y + 3, solve for y.
A X
D C
B
7
8
11
11π¦=7β8
y
11π¦=56π¦=
5611
8
15
18
Y+3 18 π¦=66
18 π¦+54=120
18 (π¦+3 )=8β15
π¦=3.667
Example: The piece of pottery shown at the right was found at an archeological site. Determine the diameter of the original piece of pottery.
1.4 in 1.4 in 0.8 in
1.4β1.4=.8 π¦
1.96=.8 π¦
2.45=π¦
ππ hπ‘ πππππππ‘ππ ππ hπ‘ πππππππππππ‘ ππ πππ’πππ‘π2.45+.8=3.25 ππ
y
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