WELCOME TO MY FAVORITE CHAPTER!! SO what is the big deal about circles? This chapter builds on your
knowledge of circles by studying some new properties of circles. Using geometric tools, you will discover many
relationships among the angles and line segments in and around circles, such as chords,tangents, secants
and arcs. You will also use coordinate geometry to describe a circle’s location on a graph and develop the
standard form of the equation of a circle. WOOOOO!!! Through this, you will better understand the concept of
a locus and how it applies to the geometry we have learned thus far.
Lesson One is about tangents. (not men at the beach). Imagine spinning then releasing a ball attached to a
string. The released ball will follow a path influenced by gravity, but its direction at the point of release is along
the line tangent to its original orbit at the point of release. The laws of physics that this obeys are related to the
geometry of this section. The tangent to a circle is related to the geometric interpretation of a derivative,
the fundamental concept of differential calculus. A derivative measures the rate of change of a function (sound
familiar) at any point and corresponds to the slope of the tangent to the graph of the function at that point.
This section is also all about circles and their parts as well as the circumference of a circle. One important
property of the circle is the relationship between the circumference and the diameter, and the area and the
radius. Circular wheels and gears are two of the most important applications of the properties of circles. Unless
you magically appeared at school today (or you walked), you arrived on a vehicle with circular wheels. Wheels
gave rise to circular gears which ushered in the industrial revolution. Potters wheels, clocks, and windmills all
based on the applications of the wheel represent great advances in civilization.
CHAPTER TEN: GOING ‘ROUND IN CIRCLES
10.1 Vocabulary
tangent, point of tangency, secant
common tangent(external and internal tangents)
circle, center, radius, diameter, congruent
circles, coplanar circles
concentric circles, circumference,
pi (π)
Essential Questions
How can parts of circles be identified and used?
How can problems involving the circumference of
a circle be solved?
What is a tangent of a circle? What properties
does it have? How is related to the radius?
What is a circumscribed polygon?
Write everything that you know about this
drawing.
ANSWERS
Lesson two becomes a bit more interesting when we see a circle’s properties through central angles, lengths
of arcs, differences between major and minor arcs, and how an arc measure is related toan arc length.
10.2 Essential Questions
How can central angles, major arcs, minor arcs,
and semicircles be identified and measured?
How can arc lengths be found?
What is the arc addition postulate and what is it
similar to?
Sketch a circle with a central angle. Identify the
length of the radius and then draw and label the
degrees of a central angle. Find the arc length for
either the major or the minor arc.
Vocabulary:
central angle, arc-minor and major, semicircle,
congruent arcs, adjacent arcs
Lesson Three speaks to the special relationship that exists between arcs and chords. You will be unpacking
these standards as you explore circles through a Geometer’s sketchpad lab. We will use the Law of Syllogism
to show that if congruent central angles have congruent chords, and congruent chords have congruent arcs,
then...You will also use the various ideas in this section to prove the theorem of analytic geometry that states
that any three non-collinear points determine a unique circle and can be used to justify a method to
constructing a circle
10.3 Essential Questions
What is the relationship between arcs and chords
and central angles in a circle?
How does a diameter of a circle relate to certain
arcs and chords of that circle?
How do you find the center of a circle if you only
have two chords in the circle?
Lesson Four is a great example of how geometry and sports are related. You already know that the angle
within which a shot will be successful becomes smaller and smaller as a shooter moves further away from a
soccer goal. With the inscribed angle theorems, you can prove that shooting positions are equivalent if they
are vertices of inscribed angles on the same circle with the same intercepted arc (goal).
10.4 VOCABULARY
inscribed angle, intercepted arc
Essential Questions
How is the measure of an inscribed angle found?
How can the measures of angles of an inscribed
polygon be found?
How does an inscribed < relate to a central < ?
If two inscribed angles intercept the same arc,
what is true?
What is true of any quadrilateral inscribed in a
circle?
If a right triangle is inscribed in a circle, what is
true?
Lesson Five and Six: One consequence of triangle similarity is the proportionality of the lengths of segments
formed by chords, secants, or tangents. The proofs in this section are a clear example of the necessity of the
similarity section that we learned previously. You will be amazed at the number of unique situations that are
formed and will probably enjoy the puzzle of interpreting angle measures formed by these lines and segments.
10.5
Essential Questions
How do similar triangles help to prove the secant,
tangent theorems?
How can measures of angles formed by lines
intersecting on or inside a circle be found? Give
all examples that we discussed in class.
How can measures of angles formed by lines
intersecting outside of a circle be found? Give all
examples that we discussed in class.
10.6 VOCABULARY
chord segment, secant segment,
external secant segment, tangent segment
Essential Questions
How can measures of segments that intersect in
the interior of a circle be found?
How can measures of segments that intersect in
the exterior of a circle be found? Show all
examples discussed in class
Lesson 8 introduces the equation for a circle. You can analyze the equation of a circle to find information that
will help you graph the circle on a coordinate plane. Once you know the coordinates of the center and the
radius of a circle, you can graph the circle, If you know just three points on a circle, you can use mathematical
ideas (such as midpoint, perpendicular bisectors, systems of equations and distance formula) to write the
equation of the circle that contains these points. This is used extensively in analytic geometry and trigonometry.
The equation of a unit circle with center at the origin leads to the fundamental trigonometric relationship that
sin2x + cos2x =1, whose importance is necessary to finding formulas for derivatives and integrals in calculus.
10.7
Essential Questions
How can the equation of a circle be written?
How can a circle be graphed on the coordinate
plane?
What is the relationship between the distance
formula and the equation of a circle in standard
form?
What is a locus as it refers to geometry?
Practice Questions
Secant segments SO and QO and
tangent segment RO intersect at
point O.
Section Number and Topic
Standards
10.1 Use Properties of
Tangents
MAFS.912.G.CO.1
10.2 Find Arc Measures
MAFS.912.G.CO.1
10.3 Apply Properties of
Chords
MAFS.912.G.C.2
10.4 Use Inscribed Angles
and Polygons
MAFS.912.G.C.3
10.5 Apply Other Angles
Relationships in Circles
MAFS.912.G.C.2
10.6 Find Segment Lengths in
Circles
MAFS.912.G.C.2
10.7 Write and Graph
Equations of Circles
MAFS.912.G.GPE.1
Learning Target
Assignment
Will use properties of a tangent to a circle. Know precise definition of angle, circle,
perpendicular line, parallel line, and line segment, based on the undefined notions
of point, line, distance along a line, and distance around a circular arc.
Will use angle measures to find arc measures.
Know precise definition of angle, circle, perpendicular line, parallel line, and line
segment, based on the undefined notions of point, line, distance along a line, and
distance around a circular arc.
Will use relationships of arcs and chords in a circle.
Identify and describe relationships among inscribed angles, radii, and chords.
Page 645: 3-12,15-18,
21,22,25-30,35,36
Challenge: 34
Page 651: 4-10 even,
12-17
Challenge: 20,25
Will use inscribed angles of circles.
Construct the inscribed and circumscribed circles of a triangle, and prove
properties of angles for a quadrilateral inscribed in a circle.
Will find the measures of angles inside or outside a circle.
Identify and describe relationships among inscribed angles, radii, and chords.
Page 666: 1-9,12-16,
18, 20-25, 29, 30
Challenge: 26
Page 675: 5,6,9-13,1518,20, Page 679: 2,7
Will find segment lengths in circles.
Identify and describe relationships among inscribed angles, radii, and chords.
Page 684: 1-3,5-7,1013,16-18
Challenge: 19
Page 692: 1-3, 8, 1418, 20,25-31, 34
Challenge: 35
Will write equations of circles in the coordinate plane.
Derive the equation of a circle of given center and radius using the Pythagorean
theorem; complete the square to find the center and radius of a circle given by an
equation.
Page 657: 3-6,9,10,1521,23,28
You will also need to complete the circles packet that can be found online on my website from
Geometry to Go.
Complete the Sketchpad lab on circles.
Construct a circle given a set of data.