Download Theorem

Warm-up
In the diagram, chords AB and CD are parallel.
Prove that AC is congruent to BD.
Theorem: In a circle, parallel chords intercept congruent arcs.
Theorem: An angle formed by two chords intersecting
inside a circle is equal to half the sum of
the intercepted arcs.
ARE = ½ (AE + DC)
Theorem: An angle formed by two secants,
two tangents, or by a secant and
a tangent drawn from a point outside
a circle is equal to half the difference
of the intercepted arcs.
APE
½
(AE
APE
AE)
APE===½
½(ANE
(AE –––AC)
CB)
P
ABC = ½ AB
Theorem: The measure of an angle formed by a
tangent and a chord of a circle is half
the measure of the arc between them.
Triangle ABC is an isosceles triangle with AB  AC.
Find the measures of x, y, and z.
A
x = 140
40
y = 70
x
z = 40
y
C
z
B
EF is tangent to circle P; AD is a diameter;
AB = 30, CD = 40, DE =50
Find the measure of each numbered angle.
E
F
6
5
9
50
2
10
4
3
A
7
B
H
1
40
C
30
8
P
D
EF is tangent to circle P; AD is a diameter;
AB = 30, CD = 40, DE =50
1 = 40
Find the measure of each numbered angle.
2 = 15
3 = 25
4 = 40
E
F
6
5
9
5 = 65
50
2
130
10
4
P
1
3
A
30
8
H
7 = 45
40
8 = 30
C
9 = 25
10 = 65
7
B
D
6 = 100
110
Part II
In the diagram, WS is tangent to
.
P.
1. Name two radii.
2. Name a diameter.
3. Name three chords.
4. Name an inscribed angle.
5. Name two right angles.
6. Name an angle congruent to R.
W
T
S
R
P
.
In the diagram, EF , CF , and
P are tangent to
.
A, and EF = 16 cm.
8
7. PB = _______
cm
4
8. AC = ______
cm
9. mACF = _______
90
8
10. FC = ______
cm
B
C
F
A
P
E
In
.
P, mWPY = 91, and mWX = 48.
43
11. mXY = _______
269
12. mWZY = _______
137
13. mZPY = ________
132
14. mWZ = _______
W
X
Y
P
Z
In the diagram, MN is tangent to the circle.
15. mLMK = _______
91
61
16. mLMN = _______
JLK
17. JMK  _______
JKL
18. JML is supplementary to  ______
N
M
122 
L
38 
J
K
Part I
1. a. Use Geometer’s Sketchpad to construct a quadrilateral inscribed
in a circle, as shown.
B and D are supplementary.
b. Make a conjecture about the relationship between the measure of B
and the measure of D.
c. Prove your conjecture.
D
The median is half the length
of the hypotenuse.
A
B
C
Theorem: Opposite angles of an inscribed quadrilateral are supplementary.
A
(Con)Cyclic Quadrilaterals
(Con)Cyclic quadrilateral is a quadrilateral
that may be inscribed in a circle
We just proved that the opposite angles of
a cyclic quadrilateral are supplementary.
The converse is also true:
If the opposite angles of a quadrilateral are
supplementary, the quadrilateral is cyclic.
This is the diagram for question 2 in the homework
In the diagram, radius AB is perpendicular to radius AC, and CD is
perpendicular to ray BD.
Are any four points in this diagram the vertices of a cyclic quadrilateral?
B
E
D
A
C
This is the diagram for question 2 in the homework
In the diagram, radius AB is perpendicular to radius AC, and CD is
perpendicular to ray BD.
Are any four points in this diagram the vertices of a cyclic quadrilateral?
B
E
D
A
C
B
2. In the diagram, radius AB is perpendicular to radius AC .
Point E is chosen randomly on minor arc BC and CD is
constructed perpendicular to ray BE at point D.
Using Geometer’s Sketchpad, make a conjecture about
the relationship between the lengths of segments CD and
DE and prove you are correct.
B
E
D
A
C
A
3. a. Use Geometer’s Sketchpad to construct a right triangle.
b. Construct the median to the hypotenuse of the right triangle.
c. Make a conjecture about the relationship between the length of the
hypotenuse and the length of the median.
The median is half the length of the hypotenuse.
Theorem: The median to the hypotenuse of a right triangle is
half the length of the hypotenuse.
Claim: A circle with
diameter AC also passes
through point B.
Suppose it doesn’t.
P
Theorem: The median to the hypotenuse of a right triangle is
half the length of the hypotenuse.
P
Theorem: The median to the hypotenuse of a right triangle is
half the length of the hypotenuse.
Theorem: When two chords intersect inside a circle, the product of the lengths of
the segments of one chord is equal to the product of the lengths of the segments
of the other chord.
(AP)(PB) = (CP)(PD)
A new twist on a familiar problem
Infinitely many rectangles with different dimensions have an area
of 36 square units. Use Geometer’s Sketchpad to construct a
rectangle whose area is 36, and which retains that area when the
dimensions are changed.
Theorem: When a tangent and a secant intersect outside a circle, the square of
the tangent’s length is equal to the product of the lengths of the secant and the
length of its external segment. (Known as The Power of the Tangent)
(AP)2 = (CP)(PD)
12
x
8
10
Theorem: When a tangent and a secant intersect outside a circle, the square of
the tangent’s length is equal to the product of the lengths of the secant and the
length of its external segment. (Known as The Power of the Tangent)
(AP)2 = (CP)(PD)
Application 1
In the diagram, PC is tangent to circle A, and secant segment PB
crosses diameter EC at point N. If BN = 11, ND = 9, and PD = 16,
1. What is the length of PC?
24
74
7
2. What is the length of the radius of circle A?
C
B
 10.57
N
11
9
A
E
D
16
P
Theorem: When a 2 secants intersect outside a circle, the product of
the length of one secant and the length of its external segment is equal to
the product of the length of the other secant and the length of its external
segment.
(AP)(PB) = (CP)(PD)
9
8
10
x
7
Simulate the following situation using Geometer’s Sketchpad.
The families living in the 3 houses shown below chipped in to buy a swing
set for their kids. They want to place the swing set so that it is the same
distance from all three houses. Where should the swing set be placed?
The circumcenter of a triangle is the center of the circumscribed circle.
It is the intersection of the perpendicular bisectors of the sides of the triangle.
The circumcenter of a triangle is equidistant from the vertices of the triangle.

Simulate the following situation using Geometer’s Sketchpad.
You are a lifeguard on a small island that is roughly shaped like a triangle.
You need to station yourself so that you are as close to each shoreline as
possible. Where should you place your chair?
The incenter of a triangle is the center of the inscribed circle.
It is the intersection of the angle bisectors of the angles of the triangle.
The incenter of a triangle is equidistant from the sides of the triangle.
The incenter of a triangle is the center of the inscribed circle.
It is the intersection of the angle bisectors of the angles of the triangle.
The incenter of a triangle is equidistant from the sides of the triangle.
The orthocenter of a triangle is the intersection of the altitudes.
The 3 altitudes of ABC
orthocenter O
O
The three perpendicular bisectors of ABC
orthocenter O
circumcenter P
O
P
The three medians of ABC
orthocenter O
circumcenter P
centroid R
O
R
P
The Nine Point Circle
orthocenter O
circumcenter P
centroid R

O


R
P

The Nine Point Circle
Midpoint of segment from
orthocenter to incenter
(center of nine point circle
Circumcenter
Euler Line