Download 10.3 Inscribed Angles

Bell work 1
Find the measure of Arc ABC, if Arc AB = 3x,
Arc BC = (x + 80º), and
__
__
AB  BC
A
C
AB = 3xº
BC = ( x + 80º )
B
Bell work 1 Answer
Since,
__
__
AB  BC , then AB  BC, thus
3x = x + 80º
2x = 80º
x = 40º = AB & BC
Therefore AB + BC =
ABC = 240º
C
A
AB = 3xº
B
BC = (x + 80º)
Bell work 2
You are standing at point X. Point X is 10
feet from the center of the circular water
tank and 8 feet from point Y. Segment XY is
tangent to the circle P at point Y. What is the
radius, r, of the circular water tank?
Y
r
P
8 ft
10 ft
•
•
X
Bell work 2
Answer
Use the Pythagorean Theorem since
segment XY is tangent to circle P at
Point Y, then it is perpendicular to the
radius, r at point Y.
r = 6 ft
Unit 3 : Circles:
10.3 Arcs and Chords
Objectives: Students will:
1. Use inscribed angles and properties
of inscribed angles to solve
problems related to circles
Words for Circles
1.
2.
3.
4.
Inscribed Angle
Intercepted Arc
Inscribed Polygons
Circumscribed
Circles
Are there any words/terms that you are
unsure of?
Inscribed Angles
Inscribed angle – is an angle whose
vertex is on the circle and whose sides
contain chords of the circle.
A
INSCRIBED ANGLE
INTERCEPTED ARC,
AB
B
Vertex on the circle
Intercepted Arc
Intercepted Arc – is the arc that lies in
the interior of the inscribed angle and
has endpoints on the angle.
A
INSCRIBED ANGLE
INTERCEPTED ARC,
AB
B
Vertex on the circle
(p. 613) Theorem 10. 8
Measure of the Inscribed Angle
The measure of an inscribed angle is equal
half of the measure of its intercept arc.
Central Angle
•
CENTER P
B
= ½ m AC
P
•
•
Inscribed angle
A
m ∕_ ABC
C
Example 1
The measure of the inscribed angle ABC = ½ the
measure of the intercepted AC.
Central Angle
•
B
m ∕_ ABC
= ½ mAC = 30º
•
30º
•
A
60º
•C
Measure of the
INTERCEPTED ARC =
the measure of the
Central Angle
AC = 60º
Example 2
Find the measure of the intercepted TU, if the
inscribed angle R is a right angle.
T
•
R
•
•
U
Example 2 Answer
The measure of the intercepted TU = 180º, if the
inscribed angle R is a right angle.
T
•
TU = 180 º
R
•
•
U
Example 3
Find the measure of the inscribed angles Q , R ,and
S, given that their common intercepted TU = 86º
Q
•
R
T
•
S
•U
TU = 86º
Example 3 Answers
Angles Q, R, and S = ½ their common intercepted arc TU
Since their intercepted Arc TU = 86º, then
Angle Q = Angle R = Angle S = 43º
Q
•
R
T
•
S
•U
TU = 86º
(p .614) Theorem 10.9
If two inscribed angles of a circle intercepted
the same arc, then the angles are congruent
Q
•
IF ∕_ Q and ∕_ S both
intercepted TU, then
T
•
∕_ Q  ∕_ S
S
•U
Inscribed vs. Circumscribed
Inscribed polygon – is when all of its
vertices lie on the circle and the
polygon is inside the circle. The Circle
then is circumscribed about the
polygon
Circumscribed circle – lies on the
outside of the inscribed polygon intersecting
all the vertices of the polygon.
Inscribed vs. Circumscribed
The Circle is circumscribed about the
polygon.
Circumscribed Circle
Inscribed Polygon
(p. 615) Theorem 10.10
If a right triangle is inscribed in a
circle, then the hypotenuse is the
diameter of the circle.
Hypotenuse
= Diameter
•
(p. 615) Converse of
Theorem 10.10
If one side of an inscribed triangle is a
diameter of the circle, then the triangle is a
right triangle and the angle opposite the
diameter is a right angle.
The triangle is
inscribed in the circle
and one of its sides
is the diameter
Angle B is a right angle
and measures 90º
•
Diameter
= Hypotenuse
B
Example
Triangle ABC is inscribed in the circle
Segment AC = the diameter of the
circle. Angle B = 3x. Find the value of x.
A
•
C
3xº
B
Answer
Since the triangle is inscribed in the circle and
one of its sides is the diameter = hypotenuse
side, then its opposite angle, Angle B, measures
90º Thus,
3x = 90º
x = 30º
(p. 615) Theorem 10.11
A quadrilateral can be inscribed in a circle iff its
opposite angles are supplementary.
X
•
•
Y
/ X + / Z = 180º, and
•
P
W
•
The Quadrilateral
WXYZ is inscribed in
the circle iff
•
/ W + / Y = 180º
Z
Example
A quadrilateral WXYZ is inscribed in circle P, if
∕_ X = 103º and ∕_ Y = 115º , Find the measures of
∕_ W = ? and ∕_ Z = ?
X
•
•
103º
Y
115º
/ X + / Z = 180º, and
•
P
W
•
The Quadrilateral
WXYZ is inscribed in
the circle iff
•
/ W + / Y = 180º
Z
Example
From Theorem 10.11
∕_ W = 180º – 115º = 65º and
∕_ Z = 180º – 103º = 77º
X
•
•
103º
Y
115º
/ X + / Z = 180º, and
•
P
W
•
The Quadrilteral WXYZ
is inscribed in the circle
iff
•
/ W + / Y = 180º
Z
Home work
PWS 10.3 A
P. 617 (9 -22) all
Journal
Write two things about “Inscribed
Angles” or “Inscribed Polygons” related
to circles from this lesson.