1. Arcs
Tangents
Angles
Sphere
Circumference
What is Unit 3 about?
We will learn to use arcs, angles, and segments in circles to solve real life problems. We will learn how to find the measure of angles related to a circle. We will find circumference and area of figures with circles. We will also find the surface and volume of spheres.

2. Crop Circles
Whether you think crop circles are made by little green men from space or by sneaky earthling geeks, you've got to admit that they are pretty dang cool...  And whoever is making them knows a ton of geometry!

3. Properties of Tangents
Tuesday, April 11, 2017
Essential Question:
How do we identify segments and lines related to circles and how do we use properties of a tangent to a circle?
Lesson 6.1
M2 Unit 3: Day 1

4. Warm Ups
1. What measure is needed to find the circumference or area of a circle?
ANSWER
radius or diameter
2. Find the radius of a circle with diameter 8 centimeters.
ANSWER
4 cm
3. A right triangle has legs with lengths 5 inches and 12 inches. Find the length of the hypotenuse.
ANSWER
13 in.
4. Solve 6x + 15 = 33.
5. Solve (x + 18)2 = x
ANSWER 7
ANSWER 3

6. Circle
The set of all points in a plane that are equidistant from a given point, called the center. E F
Circle
Tangent
Chord P
Diameter
Radius D A
Secant C B
Name of the circle: ʘ P

7. Definition
Radius – a segment from the center of the circle to any point on the circle.
Diameter – a chord that contains the center of the circle. C A B

8. Definition
Chord – a segment whose endpoints are points on the circle.

9. Definition
Secant – a line that intersects a circle in two points.

10. Definition
Tangent – a line in the plane of a circle that intersects the circle in exactly one point. P
Point of tangency O

11. A little extra information
The word tangent comes from the Latin word meaning to touch
The word secant comes from the Latin word meaning to cut.

12. EXAMPLE 1
1. Tell whether the line or segment is best described as a chord, a secant, a tangent, a diameter, or a radius.
tangent
diameter
chord
radius

13. Identify special segments and lines
EXAMPLE 2
Identify special segments and lines
2. Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of ʘ C. AC a. b. AB DE c. AE d.
SOLUTION
is a radius because C is the center and A is a point on the circle. AC a. b. AB
is a diameter because it is a chord that contains the center C. c. DE
is a tangent ray because it is contained in a line that intersects the circle at only one point. d. AE
is a secant because it is a line that intersects the circle in two points.

14. EXAMPLE 3
Find lengths in circles in a coordinate plane
3. Use the diagram to find the given lengths. a.
Radius of ʘ A b.
Diameter of ʘ A
Radius of ʘ B c.
Diameter of ʘ B d.
SOLUTION a.
The radius of ʘ A is 3 units. b.
The diameter of ʘ A is 6 units. c.
The radius of ʘ B is 2 units. d.
The diameter of ʘ B is 4 units.

15. GUIDED PRACTICE
EXAMPLE 4
4. Find the radius and diameter of ʘ C and ʘ D.
SOLUTION a.
The radius of ʘ C is 3 units. b.
The diameter of ʘ C is 6 units. c.
The radius of ʘ D is 2 units. d.
The diameter of ʘ D is 4 units.

16. Definitions
Common tangent – a line or segment that is tangent to two coplanar circles
Common internal tangent – intersects the segment that joins the centers of the two circles
Common external tangent – does not intersect the segment that joins the centers of the two circles

17. 5. Tell whether the common tangents are internal or external.
EXAMPLE 5
5. Tell whether the common tangents are internal or external. a. b.
common internal tangents
common external tangents

18. Draw common tangents
EXAMPLE 6
6. Tell how many common tangents the circles have and draw them. a. b. c.
SOLUTION a.
4 common tangents
3 common tangents b. c.
2 common tangents

19. Perpendicular Tangent Theorem 6.1
In a plane, if a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

20. Tangent Theorems
Create right triangles for problem solving.

21. Write the binomial twice. Multiply.
EXAMPLE 8
Find the radius of a circle
7. In the diagram, B is a point of tangency. Find the radius r of ʘC.
SOLUTION
You know that AB  BC , so △ ABC is a right triangle. You can use the Pythagorean Theorem.
AC2 = BC2 + AB2
Pythagorean Theorem
(r + 50)2 = r
Substitute.
(r + 50)(r + 50) = r
Write the binomial twice.
Multiply.
r2 + 50r +50r = r
r r = r
Combine Like Terms.
100r = 3900
Subtract from each side.
r = 39 ft .
Divide each side by 100.

22. Subtract from each side.
EXAMPLE 8
8. ST is tangent to ʘ Q. Find the value of r.
SOLUTION
You know from Theorem 10.1 that ST  QS , so △ QST is a right triangle. You can use the Pythagorean Theorem.
QT2 = QS2 + ST2
Pythagorean Theorem
(r + 18)2 = r
Substitute.
r2 + 36r = r
Multiply.
36r = 252
Subtract from each side.
r = 7
Divide each side by 36.

23. Perpendicular Tangent Converse
In a plane, if a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle.

24. GUIDED PRACTICE
EXAMPLE 10
9. Is DE tangent to ʘ C?
ANSWER
Yes – The length of CE is 5 because the radius is 3 and the outside portion is 2. That makes ∆CDE a Right Triangle. So DE and CD are 

25. Verify a tangent to a circle
EXAMPLE 7
10. In the diagram, PT is a radius of ʘ P. Is ST tangent to ʘ P ?
SOLUTION
Use the Converse of the Pythagorean Theorem. Because = 372, △ PST is a right triangle and ST  PT . So, ST is perpendicular to a radius of ʘ P at its endpoint on ʘ P. ST is tangent to ʘ P.

26. Congruent Tangent Segments Theorem 6.2
If two segments from the same exterior point are tangent to a circle, then they are congruent.

27. Tangent segments from the same point are
11.
RS is tangent to ʘ C at S and RT is tangent to ʘC at T.
Find the value of x.
SOLUTION
RS = RT
Tangent segments from the same point are
28 = 3x + 4
Substitute.
8 = x
Solve for x.

28. 12.

29. Homework
Page 187 # 18 – 24 all
Page 188 # 1 – 10.