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Chapter 10 - Properties of Circles

Properties of tangents, arc measures, chords, inscribed angles and polygons, angle relationships in circles, segment lengths in circles and write / graph equations of circles.
by

Christopher Cole

on 20 September 2014

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Transcript of Chapter 10 - Properties of Circles

Properties of Circles
Chapter 10.1
Use Properties of Tangents
Vocabulary
A circle is a set of all points in a plane that are equidistant from a given point called the center of the circle. A circle with center P is called "circle P" and can be written P. A segment whose endpoints are the center and any point on the circle is a radius.

A chord is a segment whose endpoints are on a circle. A diameter is a chord that contains the center of the circle.

A secant is a line that intersects a circle in two points. A tangent is a line in the plane of a circle that intersects the circle in exactly one point, the point of tangency. The tangent ray AB and the tangent segment AB are also called tangents.
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B
A
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chord
radius
diameter
secant
tangent
point of tangency
center
Chapter 10
Coplanar Circles
Two points of intersection
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1 point of intersection
(tangent circles)
no points of intersection
Common Tangents
A line, ray, or segment that is tangent to two coplanar circles is called a common tangent.
common tangents
Theorem 10.1
In a plane, a line is a tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle.
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Q
P
m
Line m is tangent to Q
if and only if line m QP.
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Theorem 10.2
Tangent segments from a common external points are congruent.
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P
T
R
S
If SR and ST are tangent segments, then SR=ST.
EXAMPLES
Prerequisite Skills
VOCABULARY CHECK
1. Two similar triangles have congruent corresponding angles and corresponding sides.

2. Two angles whose sides form two pairs of opposite rays are called .

3. The of an angle is all of the points between the sides of the angle.

SKILLS AND ALGEBRA CHECK
4. 0.6, 0.8, 0.9 5. 11, 12, 17 6. 1.5, 2, 2.5
(Use the converse of the Pythagorean theorem to classify the triangle.)
Find the value of variable.
7.
5x
(6x - 8)
o
o
o
o
o
o
8.
9.
(8x - 2)
(2x + 2)
7x
(5x + 40)
concentric
circles
Chapter 10.2
Find Arc Measures
Vocabuary
A central angle of a circle is an angle whose vertex is the center of the circle.

If m < ACB is less than 180, then the points on C that lie in the interior of <ACB form a minor arc with endpoints A and B. The points on C that do not lie on the minor arc AB form a major arc with endpoints A and B. A semicircle is an arc with endpoints that are the endpoints of a diameter.

Two arcs of the same circle are adjacent if they have a common endpoint. You can add the measures of the two adjacent arcs.

Two circles are congruent circles if they have the same radius. Two arcs are congruent arcs if they have the same measure and they are arcs of the same circle or of congruent circles. If C is congruent to D, then you can write C = D.
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minor arc AB
A
B
C
D
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major arc ADB
KEY CONCEPT
The measure of the minor arc is the measure of its central angle. The expression mAB is read as "the measure of arc AB."

The measure of the entire circle 360. The measure of a major arc is the difference between 360 and the measure of the related minor arc. The measure of a semicircle is 180.
Measuring Arcs
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A
B
D
C
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50
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mAB = 50
o
mADB = 360 - 50 = 310
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Postulate 23
Arc Addition Postulate
The measure of an arc formed by two adjactent arcs is the sum of the measures of the two arcs.
A
B
C
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mABC = + mBC
mAB
EXAMPLES
a. mAC
b. mACD
c. mADC
d. mEBD
Find the following:
mTS
mQRT
mTQ
mRST
How about some...
Chapter 10.3
Apply Properties of Chords
Theorem 10.3
In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.
A
B
C
D
AB = CD if and only if AB = CD
Theorem 10.4
If one chord is a perpendicular bisector of another chord, then the first chord is a diameter.

If QS is a perpendicular bisector of TR, then QS is a diameter of the circle.
S
T
Q
R
Theorem 10.5
If a diameter is perpendicular to a chord, then the diameter bisects the chord and its arc.

If EG is a diameter and EG DF, then HD = HF and GD = GF.
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P
E
D
G
F
H
Theorem 10.6
In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center.
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A
C
B
G
F
E
D
AB=CD if and only if EF=EG
EXAMPLES
Anyone know the...
Who can show the 3 steps needed to complete this sprinkler project? (Hint...page 665)
Famous ARCS!!!
Do you know these
How about the most famous arches? Anyone...
More people have gone to these then all the others combined!!!
Chapter 10.4
Use Inscribed Angles and Polygons
Vocabulary
An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle. The arc that lies in the interior of an inscribed angle and has endpoints on the angle is called the intercepted arc of the angle.
A polygon is an insrcibed polygon if all of its vertices lie on a circle. The circle that contains the vertices is a circumscribed circle.
circumscribed
circles
inscribed
triangle
Measure of an Inscribed
Angle Theorem
The measure of an inscribed angle is one half the measure of its intercepted arc.
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D
A
C
B
m<ADB = 1/2 mAB
Theorem 10.8
If two inscribed angles of a circle intercept the same arc, then the angles are congruent.
D
C
A
B
<ADB = <ACB
Case 2
Center C is inside the inscribed angle.
Case 1
Center C is on the inscribed angle
Case 3
Center C is outside the inscribed angle.
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C
C
C
Theorem 10.9
If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. Conversely, if one side of an inscribed triangle is a diameter of the circle, the the triangle is a right triangle and the angle opposite the diameter is the right angle.
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D
C
B
A
m<ABC = 90 if and only if AC
is a diameter of the circle.
o
Theorem 10.10
A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.

D, E, F and G lie on C if and only if m<D+m<F=m<E+m<G=180
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C
D
E
F
G
EXAMPLES
Chapter 10.5
Apply Other Angle
Relationships in Circles
Theorem 10.11
If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc.
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C
B
A
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m<1 = 1/2 mAB
m<2 = 1/2 mBCA
Intersecting Lines and Circles
Inside the circle
Outside the circle
On the circle
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Angles Inside the Circle Theorem
If two chords intersect inside a circle, then
measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.
1
2
m<1 = 1/2 (mDC+mAB)

m<2 = 1/2 (mAD+mBC)
Angles Outside the Circle Theorem
If the tangent and a secant, two tangents, or two secants intersect outside a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.
1
A
C
B
m<1 = 1/2(mBC - mAC)
2
Q
P
R
m<2 = 1/2(mPQR - mPR)
3
W
Z
X
Y
m<3 = 1/2(mXY - mWZ)
EXAMPLES
Has anyone ever actually seen the Northern Lights?
S.W.B.A.T. (Students will be able to...)
Find the measures of angles inside and outside a circle.
A
B
D
C
Chapter 10.6
Find the Segment Lengths in Circles
S.W.B.A.T. (Students will be able to...)
Find segment lengths in circles.
Vocabulary
When two chords intersect in the interior of a circle, each chord is divided into two segments that are called segments of the chord.

A secant segment is a segment that contains a chord of a circle, and has exactly one endpoint outside the circle. the part of a secant segment that is outside the circle is called the external segment.
tangent segment
external
segment
secant segment
Segments of Chords Theorem
If two chords intersect in the interior of a circle, then the products of the lengths of the segments of one chord is equal to the products of the lengths of the segments of the other chord.
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A
D
B
C
EA EB = EC ED
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Segments of Secants Theorem
If two secant segments share the same endpoint outside a circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment.
E
A
C
B
D
EA EB = EC ED
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Segments of Secants and Tangents Theorem
If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the lengths of the secant segment and the external segment equals the square of the length of the tangent segment.
E
D
C
A
EA = EC ED
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Remember the quadratic formula?
Does anyone know anything about this mission?
http://www.nasa.gov/mission_pages/cassini/main/index.html
Write and Graph Equations of Circles
Chapter 10.7
KEY CONCEPT
Standard Equation of a Circle
You can write the equation of any circle if you know the radius and the coordinates of the center.

Suppose a circle has a radius "r" and a center (h,k). Let (x,y) be a point on the circle. The distance between (x,y) and (h,k) is "r", so by the Distance Formula

(x - h) + (y-k) = r

Square both sides to find the standard equation of a circle.

(x - h) + (y - k) = r
2
2
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(h,k)
(x,y)
r
2
2
2
EXAMPLES
S.W.B.A.T.
Write equations for circles in the coordinate plane.
- Use properties of a tangent to a circle.
- Use angle measures to find arc measures.
- Use relationships of arcs and chords in a circle.
- Use inscribed angles of circles.
- Find the measures of angles inside or outside a circle.
- Find segment lengths in circles.
- Write equations of circles in the coordinate plane.
Check up.
Did you make the mark?
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