No corners or sides…Oh my!
A circle is the set of all points in a plane that are equidistant from a given point that is called the center.
They can be named by using the symbol
 and the center of the circle. The circle in the illustration below is
 C or “circle C.”
Chord - A segment whose endpoints are on a circle. Example:
Secant - A line that intersects a circle in two points. Example:
Tangent - A line in the plane of a circle that intersects the circle in exactly one point. Example:
Point of Tangency - The one point where a tangent intersects a circle.
Example:
Minor Arc - An arc less than 180 degrees. Example:
Major Arc - An arc greater than 180 degrees. Example:
Semicircle - An arc whose central angle measures 180˚.
Intercepted Arc - An arc that lies in the interior of an inscribed angle and has endpoints on the angle. Ex:
Measure of a Minor Arc - The measure of the central angle of the arc. Ex:
Measure of a Major Arc - The difference of 360˚ and the measure of the related minor arc. Ex:
Example 1:
Find the measure of arc HE.
Find the measure of arc GEF.
Find the measure of arc GFE.
Another way to measure an arc is in terms of its length. To find the length of an arc, you need to know the radius of the circle of which the arc is a part.
Arc length: If r is the radius of a circle and M is the degree measure of an arc of the circle then the length, L of the arc is given by the following:
L
2
 r

M
360
 or L

M
360

( 2
 r )

Example 2: Find the length of arc AB. a. b. c.
You Try! Find the length of arc AB. Round your answer to the nearest hundredth.
Chords and Arcs Theorem:
In a circle, or in congruent circles, the arcs of congruent chords are ____________________.
Tangent Theorem:
If a line is tangent to a circle then it is ______________________ to the radius drawn at the point of tangency.
Radius and Chord Theorem: A radius that is perpendicular to a chord of a circle
_______________ the chord.
Converse of the Tangent Theorem:
If a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is
_____________________ to the circle.
Example 1:

Example 2:
B
C
BC
Example 3:
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?
Inscribed Angle: An angle whose vertex is on a circle and whose sides contain chords of the circle.
Inscribed Angle Theorem: The measure of an angle inscribed in a circle is equal to ________________ the measure of the intercepted arc.
Example 1: Find the measure of the inscribed angle or the intercepted arc.
a. b.

You Try! Find the measure of the inscribed angle or the intercepted arc. a. b. c.
Arc-Intercept Corollary : If two inscribed angles intercept the same arc, then they have the same measure.
Example 2:
A person’s effective field of vision is about 30˚. In the diagram of the amphitheatre, a person sitting at point A can see the entire stage.
What is the measure of

B ? Can the person sitting at point B view the entire stage?
Example 3:
In

P with diameter AB, the measure of arc CB is 110˚ and the measure of arc DB is 130˚. Find the measures of

1 ,

2 ,

3 ,

4 ,

APC ,

ADB &

CAD .
You Try! Given: m

X = 130, ; m

Y = 100°
Refer to the diagram to find the measure of each of the following: a.

Z b. c.

W d.

Classification of Angles with Circles:
Case 1: Vertex is on the circle a. b.
Case 2: Vertex is inside the circle
Case 3: Vertex is outside the circle a. b. c.
Theorem: If a tangent and a secant (or a chord) intersect on a circle at the point of tangency, then the measure of the angle formed is _________ the measure of its intercepted arc.
Theorem: The measure of an angle formed by two secants or chords that intersect in the interior of a circle is _________ the _________ of the measures of the arcs intercepted by the angle and its vertex angle.
Theorem: The measure of an angle formed by two secants that intersect in the exterior of a circle is __________ the __________ of the measures of the intercepted arcs.
Example 1: Find the m

AVC in each picture.

Example 2:
Given : line TU is tangent

P at point T , measure arc
QR=90˚, measure arc RT=150˚, measure arc QS=50˚.
Find the following: a) m

STU b) c) m m


1
2
Tangent Segment:
Secant Segment:
External Secant Segment:
Chord:
Your Turn! Do activities 1, 2, and 3 on page 601-602 and use your findings to fill in the following theorems.
Theorem: If two segments are tangents to a circle from the same external point, then the segments
________________.
Theorem: If two secants intersect outside a circle, the product of the lengths of tone secant segment and its external segment equals _________________________.
Theorem: If a secant and a tangent intersect outside a circle, then the product of the lengths of the secant segment and its external segments equals ________________.
Theorem: If two chords intersect inside a circle then the product of the lengths of the segments of one chord equals ____________________.
Example 1:
Global positioning satellites are used in navigation. If the range of the satellite, AX , is 16,000 miles, what is
BX ?

Example 2:
In the figure, EX =1.31, GX =0.45, and FX =1.46. Find
HX . Round your answer to the nearest hundredth.
Example 3:
An eagle is released from captivity at a ranger station near an approximately circular lake, and it builds a nest in a tree opposite the shoes. The distance from the station to the lake is 300 yd along a road tangent to the lake, and 50yd along a line straight to the nest. How far is it from the nest to the ranger station?
Example 4:
In the figure, AX = 0.26, XC = 0.91 and DX = 0.27. Find XB.
Example 1:
Given x 2 + y 2 = 25, sketch and describe the graph by finding ordered pairs that satisfy the equation. If you have a graphing calculator graph to check.

You Try!
Sketch the graph of x
2
2 + y
2
= 49
Deriving the Equation of a Circle:
Moving the Center of a Circle:
To find the standard form of the equation of a circle centered at a point (h, k) that is not at the origin the equation is a little more complicated:
( x – h )
2
+ ( y – k )
2
= r
2
It is derived from the diagram below:
Example 2: Given (x – 7)
2
+ (y + 3)
2
= 36, find the center of the circle and its radius.
You Try! For each equation below find the radius and the center of the circle. a. (x + 3)
2
+ (y – 3)
2
= 49 b. (x – 3)
2
+ (y + 3)
2
= 49 c. (x – 4) 2 + (y – 5) 2 = 64 d. (x + 2) 2 + (y – 5) 2 = 50