1. Lesson 5
Circles

3. Definitions
A circle is the set of all points in a plane that are the same distance from a fixed point called the center of the circle.
A radius of a circle is a line segment extending from the center to the circle.
A diameter is a line segment that joins two points on the circle and passes through the center.
radius
center
diameter

4. Naming a Circle
A circle in a diagram is named by its center. The circle at right is called circle O or:
If there is more than one circle in a diagram with the same center, this notation does not suffice.
Note: two circles in the same plane with the same center are called concentric circles. O

5. The word radius (plural: radii) is also used to denote the length of a radius (all radii have the same length).
The word diameter is also used to denote the length of a diameter (all diameters have the same length).
Note that the diameter of a circle is twice its radius.

6. Chords A chord is any line segment that joins two points on a circle.
Therefore, a diameter is an example of a chord. It is the longest possible chord.

7. Chords and Radii
Given a chord in a circle, any radius that bisects the chord (passes through its midpoint) is perpendicular to that chord.
Also, if a radius is perpendicular to a chord, then it bisects the chord.

8. Distance to Chords
The distance from the center of a circle to a chord is measured along the radius that is perpendicular to the chord.
Chords that are the same distance from the center are the same length.
Also, chords that are the same length are the same distance from the center.

9. Example
In a circle of radius 5, a chord has length 8. Find the distance to the chord from the center of the circle.
Let C be the center of the circle and A and B the endpoints of the chord. Let M be the midpoint of the chord so that
Then
So, by the Pythagorean Theorem, B M 4 A C 5

10. Example In the figure, What is
Since are both radii, they are congruent.
So, angles A and B are congruent too. Therefore, P A B

11. Intersecting Chords
Consider the two intersecting chords in the figure. Each chord is cut into two pieces.
The product of the lengths of the two pieces on one chord equals the product of the lengths of the two pieces on the other chord: A B C D P

12. Example In the figure, If then find Let denote the length of CP.
Then is the length of PD.
Then A B C D P 3
11-x 11 x 6

13. Tangents
Given a circle, a line is tangent to the circle if it touches it just once. Such a line is called a tangent or a tangent line. The point where the tangent touches the circle is called the point of tangency.
We can also speak of tangent segments or rays.
One crucial property of tangents is that the radius drawn to the point of tangency is perpendicular to the tangent.

14. Two Tangents P A B
Given a point P outside of a circle, there are two lines passing through P that are tangent to the circle.
Also, the distance from P to each point of tangency is the same:

15. Example
From a point P outside of circle Q, draw a tangent to the point of tangency A.
If then what is the radius of circle Q?
Draw radius Note that is a right angle. So, by the Pythagorean Theorem: P 12 15 A Q

16. Secants
Given a circle, a line that intersects the circle twice is called a secant line or a secant.
Line segments and rays may also be secants.
Often, secants are drawn from a point outside of the circle to a point on the circle.

17. Two Secants P A B C D
From a point P outside of a circle, draw two secants as in the figure.
Then
In words: the outside part times the whole equals the outside part times the whole.

18. A Tangent and a Secant P A C D
From a point P outside a circle, draw a tangent and a secant as in the figure.
Then
Note that we can remember the same words from the last slide: the outside part times the whole equals the outside part times the whole. But this time, the outside part and the whole of the tangent are the same.

19. Example In the figure, Find Let represent the length of PA. Then P 4 xB C D
In the figure,
Find
Let represent the length of PA. Then 4 x 7 6

20. Example In the figure, is tangent to circle Q.
If and then what is the radius of circle Q?
Let r denote the radius. Then: T 6 3 S r r P Q R

21. Arcs An arc is an unbroken part of a circle.
For example, in the figure, the part of the circle shaded red is an arc.
A semicircle is an arc equal to half a circle.
A minor arc is smaller than a semicircle.
A major arc is larger than a semicircle.

22. Naming Arcs
A minor arc, like the one in red in the figure, can be named by drawing an arc symbol over its endpoints:
Sometimes, to avoid confusion, a third point between the endpoints is used to name the arc:
The reason for this is to avoid ambiguity because, given two points on a circle, there are two arcs between them (the long way around the circle or the short way). A P B

23. The arc highlighted in red in the figure would be called
Semicircles and major arcs must be named with three (or sometimes more) points.
The arc highlighted in red in the figure would be called
It appears to be a major arc.
If we wrote then we would be referring to the part of the circle that is not highlighted in red (a minor arc, it seems). A B C

24. The Measure of an Arc
Each arc has a degree measure between 0 degrees and 360 degrees.
A full circle is 360 degrees, a semicircle is 180 degrees, a minor arc measures less than 180 degrees, and a major arc measures more than 180 degrees.
If an arc is a certain fraction of a circle, then its measure is the same fraction of 360 degrees. Some sample arc measures are given below. C A E B D F G H

25. Example In the figure, and Find
Let denote the measure of each of the two equal arcs.
Then P Q R

26. Central Angles
Given a circle, a central angle is an angle whose vertex is at the center of the circle.
In the figure, the center of the circle is and is a central angle that intercepts arc
The measure of a central angle is equal to the measure of the arc it intercepts. P A Q B

27. Inscribed Angles
In a circle, an inscribed angle is an angle whose vertex is on the circle and whose sides are chords.
In the figure, is an inscribed angle and it intercepts arc
The measure of an inscribed angle is half the measure of its intercepted arc. P A B Q

28. Angle Inscribed in a Semicircle
An important example of an inscribed angle is one that intercepts a semicircle.
In the figure, is a diameter of the circle which divides the circle into two semicircles.
intercepts a semicircle and since a semicircle measures the measure of the inscribed angle is A B P

29. A Special Kind of Inscribed Angle
In the figure, is tangent to the circle.
In this case, is called an inscribed angle and it intercepts arc
And, T A P B

30. Example In the figure, P is the center of the circle and Find and C B

31. Example In the figure, O is the center of the circle, and Find A D O PB C

32. Angles with Vertex Outside a Circle
Given a circle, suppose point P is outside the circle and two secants are drawn from P as in the figure.
Then intercepts two arcs:
This angle and these arcs are related by the formula: P A B C D

33. More Angles with Vertex Outside a Circle
The formula on the previous slide holds even if one or both sides of the angle are tangents instead of secants.
In the figure, are tangents. Then: P S T A B

34. Angles with Vertex Inside a Circle
Consider in the figure. This angle intercepts Also, if you extend the sides of backwards then they intercept
This angle and these arcs are related by the formula: A B C D P

35. Example In the figure, and Find
Let denote the measures of respectively.
Note A B C D P