1. Tangents to Circles

2. Theorem: Two chords are congruent IFF they are equidistant from the center.B
AD  BC
IFF
LP  PM A M P L C D

3. Ex. 1: IN A, PR = 2x + 5 and QR = 3x –27. Find x.

4. Ex. 2: IN K, K is the midpoint of RE
Ex. 2: IN K, K is the midpoint of RE. If TY = -3x + 56 and US = 4x, find x. U T K E R S Y
x = 8

5. 3.) Find the length of CV

6. 2 Facts about Tangents

7. Fact #1
A tangent line is ALWAYS perpendicular to the radius of the circle drawn to the point of tangency.
radius
90 degrees = perpendicular
tangent

8. 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 angle
will always be the one
on the outside of the
circle
radius
tangent

9. Example – Find the length of AC
a2 + b2 = c2
= c2
= c2
89 = c2
= c

10. Example – find x ANSWER: x = 8
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
a2 + b2 = c2
= c2
= c2
169 = c2
13 = c ?
This whole length is 13.
x + 5 = 13
x = 8 5 12
ANSWER: x = 8

11. Example Find KY a2 + b2 = c2 102 + b2 = 242 100 + b2 = 576 476 = b2

12. Example Does this picture show a tangent? a2 + b2 = c2
It must satisfy Pythagorean Theorem
a2 + b2 = c2
= (18+7)2
625 = 625
Yes!

13. Fact #2
If two segments from the same exterior point are tangent to a circle, then they are congruent.
tangent #1
exterior point
tangent #2
They are congruent.

14. 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

15. Example
exterior point
Because of Fact #2, x=14.

16. Example
Find length of tangent

17. Find NP N T S P R Q