special segments in a circle find measures of segments that intersect in the interior of a circle....
TRANSCRIPT
Special Segments in a Circle• Find measures of segments that intersect in the interior of
a circle.• Find measures of segments that intersect in the exterior of
a circle.
A Tibetan Mandala exhibiting a six-pointed star.
SEGMENTS INTERSECTING INSIDE A CIRCLE
1) Construct two intersecting chords in a circle.
3) Draw PS and RQ.
2) Name the chords PQ and RS intersecting at T.
S
P
Q
R
T
SEGMENTS INTERSECTING INSIDE A CIRCLE
Analyze:
PTS RTQVertical Angles
S
P
Q
R
T
P RAngles intercept the same arc
By angle-angle similarity,
TQST
RTPT or PT ∙ TQ = RT ∙ ST
S
P
Q
R
T
Theorem
If two chords intersect in a circle, then the products of the measures of the segments of the chords are equal.
TQST
RTPT or PT ∙ TQ = RT ∙ ST
C
A
B
D
E
Example 1 Intersection of Two Chords
3
4x
6
Find x
Example 2 Intersection of Two Chords
Find x
x
812
9
Example 3 Solve Problems
What is the radius of the circle containing the arc if the arc is not a semicircle? 24 24
12
Example 3 continued Solve Problems
Solution:
24 x 24 = 12x
576 = 12x
48 = x
x
Diameter = 48 + 12= 60
What is the radius of the circle containing the arc if the arc is not a semicircle?
Radius = 60/2 = 30
24 24
12
SEGMENTS INTERSECTING OUTSIDE A CIRCLE
Theorem
E
BC
A
D
If two secant segments are drawn to a circle from an exterior point, then the product of the measures of one secant segment and its external secant segment is equal to the product of the measures of the other secant segment and its external secant segment.
ADAEACAB
Example 4 Intersection of Two Secants
12 2
3 S
PQ
R
Find RS if PQ = 12, QR = 2, and TS = 3.
T
Let RS = xx
)4)(7(0
2830
328
)3()212(2
2
2
xx
xx
xx
xx
RTRSPRQR
7
07
x
x
4
04
x
x
Disregard the negative value
Example 5 Intersection of Two Secants
Find x if EF = 10, EH = 8, and FG = 24.
G
F IH
Ex
8
10
24
5.34
2768
340864
)2410(10)8(8
x
x
x
x
Theorem
If a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment.
WYWZWXWX Z
WX
Y
Example 6 Intersection of a Secantand a Tangent
Find x.
4
C
B
A
D
x + 2 x
80
16220
2216
)22(16
)2(44
2
2
2
xx
xx
xx
xx
xxx
The expression is not factorable. Use the quadratic formula.
aacbb
x2
42
37.241322
)2(2
)16)(2(422 2
x
or41322
Disregard the negative solution
Example 7 Intersection of a Secantand a Tangent
Find x.
x + 2
x
x + 4)2)(8(0
1660
22168
)22(168
)2()4)(4(
2
22
2
xx
xx
xxxx
xxxx
xxxxx
8
08
x
x
2
02
x
x
Disregard the negative value