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

𝑐2=42+42

𝑐2=32𝑐=√32=4√2

105

102=52+𝑥2

𝑥2=75

𝑥=√75=5√3h𝑙𝑒𝑛𝑔𝑡 𝑜𝑓 h𝑐 𝑜𝑟𝑑=10 √3

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

11𝑦=7∗8

y

11𝑦=56𝑦=

5611

8

15

18

Y+3 18 𝑦=66

18 𝑦+54=120

18 (𝑦+3 )=8∗15

𝑦=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

1.4∗1.4=.8 𝑦

1.96=.8 𝑦

2.45=𝑦

𝑆𝑜 h𝑡 𝑒𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝑜𝑓 h𝑡 𝑒𝑜𝑟𝑔𝑖𝑛𝑎𝑙𝑝𝑜𝑡 𝑖𝑠 𝑒𝑞𝑢𝑎𝑙𝑡𝑜2.45+.8=3.25 𝑖𝑛

y