Download Identifying and Describing, Parts of Circles

Unit 4.3
Identifying, Describing, and
Applying Theorems about Circles
Understand and apply theorems about circles
G-C.2, Identify and describe relationships among inscribed angles, radii, and chords.
Include the relationship between central, inscribed, and circumscribed angles; inscribed
angles on a diameter are right angles; the radius of a circle is perpendicular to the
tangent where the radius intersects the circle.
G-C.3 Construct the inscribed and circumscribed circles of a triangle, and prove
properties of angles for a quadrilateral inscribed in a circle.
Expressing Geometric Properties with Equations G-GPE
Use coordinates to prove simple geometric theorems algebraically [Include
distance formula; relate to Pythagorean theorem]
G-GPE.4 Use coordinates to prove simple geometric theorems algebraically. For
example, prove or disprove that a figure defined by four given points in the coordinate
plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered
at the origin and containing the point (0, 2).
Congruence
G-CO
Make geometric constructions [Formalize and explain processes]
G-CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in
a circle.
Standards of Math Practice
 SMP 4 Model with Mathematics
 Construct, inscribe, and circumscribe polygons
 SMP 5 Use Appropriate Tools Strategically
 Use technology and construction tools
appropriately
Essential Questions


What is the relationship between a radii and a chord?
Where would you see each part of a circle in the real
world?
Identifying and Describing,
Parts of Circles
G-C.2 Identify and describe relationships among inscribed angles,
radii, and chords. Include the relationship between central,
inscribed, and circumscribed angles; inscribed angles on a
diameter are right angles; the radius of a circle is perpendicular
to the tangent where the radius intersects the circle.
Review Concepts
 A circle is a set of points in a plane that are the same distance from
a given point called the center.
The radius is the distance from the
center to any point on the outside of
the circle.
A
C
Q
QB is a radius
B
The diameter of a circle is the
distance across a circle.
AE is a diameter
D
E
A diameter is also a chord.
A chord is a line segment with
endpoints on the circle
CD is a chord
Angles of a Circle
 <ABC is an inscribed angle. An inscribed angle has its vertex
on the circle and with sides that are chords of the circle.
 AC is an inscribed arc. An inscribed arc is the arc that lies in
interior of an inscribed angle and has its endpoints on the
angle.
<AOB is a central angle. A
central angle has its vertex
at the center and sides on
the circle.
Relationship between Angles and
Arcs
 A central angle has the same measure as its
intercepted arc.
 An inscribed angle is half the measure of its
intercepted arc.
Relationship between Angles and Arcs
(cont)
 It follows that an
inscribed angle with rays
that are on endpoints of
the diameter are right
angles
Tangent and Secant Lines
 A tangent line intersects a circle at exactly one point.
 A secant line intersects a circle at two points
What is the
difference
between a secant line
and a chord??
Tangent Lines and Radii
 The radius of a circle and the tangent line form a right
angle at the point the tangent line intersects the
circle.
Circumscribed Angles
 A circumscribed angle is an angle that is formed by
two tangent lines.
• The tangent lines and the radii
form a quadrilateral.
• <J and <S are right angles
because they are at the
intersection of the tangent line
and radii.
• Because the sum of the angles
of a quadrilateral is 360 degrees
• m<JOS + m<SRJ = 180
• A circumscribed angle and its
corresponding central angle are
supplementary.
Circumscribed Angles
(cont)
 The measure of a circumscribed angle is half the
difference of the two arc measures it intersects.
Angles formed by Chords
 When two chords intersect "inside" a circle, four
angles are formed. At the point of intersection, two
sets of vertical angles can be seen in the corners of
the X that is formed on the
picture. Remember: vertical angles are equal.
 The measure of x is equal to half the sum
of the intercepted arcs.
X = ½ (mAC + mBD)
Inscribed Quadrilaterals
A Cyclic Quadrilateral's opposite
angles add to 180°:
•a + c = 180°
•b + d = 180°
Measure of arcWXY = 138
The rest of the circle is arc WZY.
The entire circle is 360 therefore
mWZY = 360 – 138 = 222
The inscribed angle of arc WZY is
<WXY.
m<WXY = ½ (222) = 111
111 + 69 = 180