1. Properties of Tangents
Section 11.2
Properties of Tangents

2. Goal
Use properties of a tangent to a circle.

3. Key Vocabulary
Tangent segment

4. Theorems 11.1 Perpendicular Tangent Theorem
11.2 Perpendicular Tangent Converse
11.3 Congruent Tangent Segments

5. Tangents
A tangent is a line in the plane of a circle that intersects the circle in exactly one point. Line j is a tangent. j

6. Theorem 11.1 Perpendicular Tangent Theorem
Words; If a line is tangent to a circle then it is perpendicular to the radius drawn at the point of tangency.. B ℓ C
Symbols; If ℓ is tangent to ⊙ C at B, then ℓ ⊥

7. Theorem 11.2 Perpendicular Tangent Converse
Words; In a plane, if a line is perpendicular to a radius of a circle at its endpoint on a circle, then the line is tangent to the circle. B ℓ C
Symbols; If ℓ ⊥ , then ℓ is tangent to ⊙ C at B.

8. Example 1 AC
is tangent to B at point C. Find BC.
SOLUTION
is a radius of B, so you can apply Theorem 11.1 to conclude that and are perpendicular. BC AC
So, BCA is a right angle, and ∆BCA is a right triangle. To find BC, use the Pythagorean Theorem.

9. Subtract 144 from each side. Find the positive square root.
Example 1
(BA)2 = (BC)2 + (AC)2
Pythagorean Theorem
132 = (BC)
Substitute 13 for BA and 12 for AC.
169 = (BC)
Multiply.
25 = (BC)2
Subtract 144 from each side.
5 = BC
Find the positive square root.

10. Example 2
You are standing at C, 8 feet from a silo. The distance to a point of tangency is 16 feet. What is the radius of the silo?
SOLUTION
Tangent is perpendicular to radius at B, so ∆ABC is a right triangle. So, you can use the Pythagorean Theorem. AB BC

11. The radius of the silo is 12 feet.
Example 2
(AC)2 = (AB)2 + (BC)2
Pythagorean Theorem
(r + 8)2 = r
Substitute r + 8 for AC, r for AB,
and 16 for BC.
r2 + 16r + 64 = r
(r + 8)(r + 8) = r2 + 16r + 64
16r + 64 = 256
Subtract r2 from each side.
16r = 192
Subtract 64 from each side.
r =12
Divide each side by 16.
ANSWER
The radius of the silo is 12 feet.

12. How can you show that must be tangent to D? EF
Example 3
Verify a Tangent to a Circle
How can you show that must be tangent to D? EF
SOLUTION
Use the Converse of the Pythagorean Theorem to determine whether ∆DEF is a right triangle.
Compare (DF)2 with (DE)2 + (EF)2.
(DF) (DE)2 + (EF)2 = ?
Substitute 15 for DF, 9 for DE, and 12 for EF. = ?
Multiply. = ?
225 = 225
Simplify.

13. Example 3
Verify a Tangent to a Circle
∆DEF is a right triangle with right angle E. So, is perpendicular to By Theorem 11.2, it follows
that is tangent to D. EF DE

14. Example 4: ALGEBRA is tangent to at point R. Find y.
Because the radius is perpendicular to the tangent at the point of tangency, This makes a right angle and  a right triangle. Use the Pythagorean Theorem to find QR, which is one-half the length y.

15. Example 4: Pythagorean Theorem Simplify. Subtract 256 from each side.
Take the square root of each side.
Because y is the length of the diameter, ignore the negative result.
Answer: Thus, y is twice

16. Your Turn:
is a tangent to at point D. Find a.
Answer: 15

17. Example 5a: Determine whether is tangent to
First determine whether ABC is a right triangle by using the converse of the Pythagorean Theorem.

18. Example 5a: Pythagorean Theorem Simplify.
Because the converse of the Pythagorean Theorem did not prove true in this case, ABC is not a right triangle.
Answer: So, is not tangent to

19. Example 5b: Determine whether is tangent to
First determine whether EWD is a right triangle by using the converse of the Pythagorean Theorem.

20. Example 5b: Pythagorean Theorem Simplify.
Because the converse of the Pythagorean Theorem is true, EWD is a right triangle and EWD is a right angle.
Answer: Thus, making a tangent to

21. Your Turn:
a. Determine whether is tangent to
Answer: yes

22. Your Turn:
b. Determine whether is tangent to
Answer: no

23. Tangent Segments
A Tangent Segment touches a circle at one of the segment’s endpoints and lies in the line that is tangent to the circle at that point. C A B
Tangent Segment

24. Theorem 11.3 Congruent Tangent Segments
Words; If two segments from the same point outside a circle are tangent to the circle, then they are congruent. R S T
Symbols;

25. Example 6 Find the value of x. is tangent to C at B.
is tangent to C at D. AB AD
SOLUTION
AD = AB
Two tangent segments from
the same point are congruent.
2x + 3 = 11
Substitute 2x + 3 for AD and 11 for AB.
2x = 8
Subtract 3 from each side.
x = 4
Divide each side by 2.

26. and are tangent to A. Find the value of x.
Your Turn:
and are tangent to A. Find the value of x. CB CD 1. 15
ANSWER 2.
ANSWER 3

27. Example 7:
ALGEBRA Find x. Assume that segments that appear tangent to circles are tangent.
are drawn from the same exterior point and are tangent to so are drawn from the same exterior point and are tangent to

28. Example 7: Definition of congruent segments Substitution.
Use the value of y to find x.
Definition of congruent segments
Substitution
Simplify.
Subtract 14 from each side.
Answer: 1

29. Your Turn:
ALGEBRA Find a. Assume that segments that appear tangent to circles are tangent.
Answer: –6

30. Example 8:
Triangle HJK is circumscribed about Find the perimeter of HJK if

31. Example 8: Use Theorem 10.10 to determine the equal measures.
We are given that
Definition of perimeter
Substitution
Answer: The perimeter of HJK is 158 units.

32. Your Turn:
Triangle NOT is circumscribed about Find the perimeter of NOT if
Answer: 172 units

33. Assignment
Pg. 598 – 600; #1 – 39 odd