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