tangents to circles
DESCRIPTION
Tangents to Circles . Theorem: Two chords are congruent IFF they are equidistant from the center. B. AD BC IFF LP PM. A. M. P. L. C. D. Ex. 1: IN A, PR = 2x + 5 and QR = 3x –27. Find x. R. x. x. A. P. Q. x = 32. - PowerPoint PPT PresentationTRANSCRIPT
Tangents to Circles
Theorem: Two chords are congruent IFF they are equidistant from the center.
A
B
C
D
M
L
P
AD BCIFFLP PM
Ex. 1: IN A, PR = 2x + 5 and QR = 3x –27. Find x.
P
R
Q
A
xx
x = 32
Ex. 2: IN K, K is the midpoint of RE. If TY = -3x + 56 and US = 4x, find x.
Y
T
S
K
x = 8
U
RE
3.) Find the length of CV
2 Facts about Tangents
Fact #1
• A tangent line is ALWAYS perpendicular to the radius of the circle drawn to the point of tangency.
tangent
radius90 degrees = perpendicular
What this fact means….
• What this means is that you can make a right triangle and use the pythagorean theorem to find distances.
• The right anglewill always be the oneon the outside of the circle
tangent
radius
Example – Find the length of AC
a2 + b2 = c2
52 + 82 = c2
25 + 64 = c2
89 = c2
= c89
Example – find x
Since a radius of the circle is 5, any radius is 5…
Since it is a radius drawn to a point of tangency, it is perpendicular to the tangent.5
5
12
? a2 + b2 = c2
122 + 52 = c2
144 + 25 = c2
169 = c2
13 = c
This whole length is 13.x + 5 = 13x = 8
ANSWER: x = 8
Example
• Find KYa2 + b2 = c2
102 + b2 = 242
100 + b2 = 576476 = b2
= b2 119
Example
• Does this picture show a tangent?
• It must satisfy Pythagorean Theorem
a2 + b2 = c2
72 + 242 = (18+7)2
625 = 625Yes!
Fact #2
• If two segments from the same exterior point are tangent to a circle, then they are congruent.
exterior point
tangent #1
tangent #2
They are congruent.
What this fact means….• What this means is that you can set the 2
tangents equal to each other• Tangent 1 = tangent 2
tangent #1
tangent #2
Example
Because of Fact #2, x=14.
exterior point
Example
• Find length of tangent
T S
Q10 4
18NP
P
N
R
12