Content Area
Standard
Strand
Math
4.HS High School
G-C. Geometry – Circles
Content Statement
CPI#
Cumulative Progress Indicator (CPI)
ACSSSD
Objectives
Understand and apply theorems about circles
4.HS.G-C.1 Prove that all circles are similar.
1. Demonstrate an understanding that
a circle is the set of all points in a
plane that are equidistant from a
given point in that plane which is
the center of the circle.
2. Demonstrate an understanding that
a radius is a line segment drawn
from the center to any point on the
circle.
3. Demonstrate an understanding that
circumference of a circle is found
using the equation C=2r.
4. Demonstrate an understanding that
a chord is a line segment drawn
from any point on the circle to
another point on the circle.
5. Demonstrate an understanding that
the diameter is a chord that is
drawn through the center of the
circle.
6. Demonstrate and understanding
that an arc is an unbroken part of
the circle between any two points,
that those two points are known as
endpoints, and that endpoints
divide the circle into two arcs.
7. Demonstrate an understanding that
semicircles are arcs that have the
same endpoints as a diameter and
are named by the endpoints and
another point on the arc.
8. Demonstrate an understanding that
a major arc is an arc that is longer
than a semicircle and named by its
endpoints and another point on the
arc.
9. Demonstrate an understanding that
a minor arc is an arc that is shorter
than a semicircle and is named by
its endpoints.
10. Demonstrate an understanding that
central angles are angles in the
plane of a circle whose vertices are
located at the center of the circle
and whose sides intersect the circle;
and that central angles are used to
find the measures of arcs.
11. Demonstrate an understanding that
an intercepted arc of a central angle
lies within the two endpoints that
form the sides of the central angle
and contains all the points within
the interior of the central angle.
12. Demonstrate an understanding that
arcs are measured in terms of
degrees; degree measures of minor
arcs are the measures of their
central angles and the degree
measures of major arcs are 360°
minus the degree measures of there
minor arcs.
13. Demonstrate an understanding that
arc length can be found by using
the equation L=M/360°(2r) where
M=degree measure of the arc.
14. Demonstrate an understanding of
Chords and Arcs Theorem and its
converse that states that in the same
circle or in congruent circles,
congruent chords have congruent
arcs and congruent arcs have
congruent chords.
15. Demonstrate an understanding that
if two chords are equidistant from
the center of a circle then they and
their arcs are congruent.
16. Demonstrate an understanding of
triangle congruency (SSS, SAS)
used to prove chords and arcs
theorem, that the radii drawn from
the center of the circle to the
endpoints of the chord or arc form
the sides of the congruent triangles,
and the central angles form the
congruent angles.
4.HS.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.
1. Demonstrate an understanding that
an inscribed angle is an angle
whose vertex is on the
circumference of a circle and
whose sides are chords of the
circle, and that the arc located
within the inscribed angle is the
intercepted arc.
2. Demonstrate an understanding of
the Inscribed Angle Theorem,
which states that the measure of
an inscribed angle is equal to half
the measure of its intercepted
arc. The measure of an
intercepted arc, which is equal to
its central angle, is twice the
3.
4.
5.
6.
7.
8.
4.HS.G-C.3 Construct the inscribed and
circumscribed circles of a triangle, and
prove properties of angles for a
quadrilateral inscribed in a circle.
measure of the inscribed angle.
Demonstrate an understanding of
arc-intercept corollary, which
states that inscribed angles on
the same arc of a chord are equal
since they intercept the same arc.
Demonstrate an understanding of
right-angle corollary, which
states that an inscribed angle that
intercepts a semicircle is 90° or a
right angle.
Identify a circumscribed angle as
an angle whose rays are tangent
to the circle, whose vertex is
outside the circle and is equal to
180° minus the measure of the
central angle that intercepts the
same arc.
Identify a tangent as a line that
intersects a circle at exactly one
point, called the point of tangency.
Identify a secant as line that
intersects a circle at two points.
Demonstrate an understanding of
tangent theorem, which states that a
tangent is perpendicular to the
radius drawn to the point of
tangency and the converse of the
tangent theorem, which states that a
line perpendicular to a radius at its
endpoint on a circle is a tangent.
1. Construct angle bisectors.
2. Construct perpendicular bisectors.
3. Construct the inscribed circle of a
triangle.
4. Construct the circumscribed circle
about a triangle.
5. Prove that opposite angles of an
inscribed quadrilateral are
supplementary.
4.HS.G-C.4 Construct a tangent line from a point
outside a given circle to the circle.
1. Two tangent lines to a circle can be
constructed using a given point
outside of the circle.
2. Construct a tangent line to a circle
from a given point outside the
circle using a variety of tools.
Find arc lengths and areas of sectors of circles
4.HS.G-C.5 Derive using similarity the fact that the 1. Demonstrate an understanding that
length of the arc intercepted by an angle
arc length is a fraction of the
is proportional to the radius, and define
circumference of the circle, and
the radian measure of the angle as the
that fraction is found by dividing
constant of proportionality; derive the
the measure of the central angle by
formula for the area of a sector.
360 degrees.
2. Demonstrate an understanding that
1 degree is equal to /180 radians.
3. Demonstrate an understanding that
arc length is proportional to the
radius of the circle and the constant
of proportionality is the radian
measure of the angle.
4. Demonstrate an understanding that
area of a sector is a fraction of the
area of the circle and that fraction
is found by dividing the measure of
the central angle by 360 degrees.
5. Calculate the length of an
intercepted arc.
6. Demonstrate that the constant of
proportionality between arc length
and the radius of the circle is the
radian measure of the central angle.
7. Derive the formula for the area of a
sector using similarity.
8. Calculate the area of a sector.