Download Prove geometric theorems - Township of Union Public Schools

TOWNSHIP OF UNION PUBLIC SCHOOLS
MA 280 Honors Geometry
Curriculum Guide
2012
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Board Members
See web site: twpunionschools.org
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TOWNSHIP OF UNION PUBLIC SCHOOLS
Administration
See web site: twpunionschools.org
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DEPARTMENT SUPERVISORS
See web site: twpunionschools.org
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Honors Geometry Curriculum Committee
Steven Chaneski – Teacher of Mathematics
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Table of Contents
Title Page
1.
Board Members/Administration/Department Supervisors/Curriculum Committee
2.
Table of Contents
3.
District Mission/Philosophy Statement/District Goals
4
Course Description/Recommended Texts
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Course Proficiencies
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Pacing Guide
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Curriculum Units
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Appendix: New Jersey Core Curriculum Content Standards
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Mission Statement
The Township of Union board of Education believes that every child is entitled to an education, designed to meet his or her individual
needs, in an environment that is conductive to learning. State standards, federal and state mandates, and local goals and objectives,
along with community input, must be reviewed and evaluated on a regular basis to ensure that an atmosphere of learning is both
encouraged and implemented. Furthermore, any disruption to or interference with a healthy and safe educational environment must be
addressed, corrected, or when necessary removed in order for the district to maintain the appropriate educational setting.
Philosophy Statement
The Township of Union Public School District, as a societal agency, reflects democratic ideals and concepts through its educational
practices. It is the belief of the Board of Education that a primary function of the Township of Union Public School System is
formulation of a learning climate conductive to the needs of all students in general, providing therein for individual differences. The
school operates as a partner with the home and community.
Statement of District Goals
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Develop reading, writing, speaking, listening, and mathematical skills.
Develop a pride in work and a feeling of self-worth, self-reliance, and self discipline.
Acquire and use the skills and habits involved in critical and constructive thinking.
Develop a code of behavior based on moral and ethical principals.
To be able to work with others cooperatively.
Acquire a knowledge and appreciation of the historical record of human achievement and failures and current societal issues.
Acquire a knowledge and understanding of the physical and biological sciences.
Efficient and effective participation in economic life and the development of skills to enter a specific field of work.
Appreciate and understand literature, art, music, and other cultural activities.
Develop an understanding of the historical and cultural heritage.
Develop a concern for the proper use and/or preservation of natural resources.
Develop basic skills in sports and other forms of recreation.
Course Description
Geometry
Geometry students will focus on reasoning about two and three dimensional figures and their properties. They will develop facility
with a broad range of ways of representing geometric ideas, including coordinates, transformations and vectors. Students will be
making conjectures, proving theorems, and finding counterexamples to refute false claims. This course builds on geometry students
encountered in middle school and extends some concepts addressed in Algebra I. Students will be able to understand how geometry
relates to the real world and how often math is used in their lives today and in the future.
Curriculum Units
Unit 1:
Unit 2:
Unit 3:
Unit 4:
Unit 5:
Geometric Representations
Reasoning in Geometric Situations
Congruence, Similarity, and Right Triangle Trigonometry
Circles
Three – Dimensional Geometry
Recommended Texts
Geometry, Ron Larson, McDougal Littell, 2007
Flatland, Edwin A. Abbott, Dover Thrift Edition, 1992
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Course Proficiencies
Unit 1: Geometric Representations:
SWBAT:
- Understand and solve problems using geometric representations.
- Show position and represent motion in the coordinate plane using vectors
Unit 2: Reasoning in Geometric Situations:
SWBAT:
- Use the laws of inductive and deductive reasoning to draw conclusions.
- Prove geometric theorems and statements using formal proofs.
- Apply properties of parallel and perpendicular lines.
- Understand and apply properties of triangles
- Make geometric constructions.
Unit 3: Congruence, Similarity, and Right Triangle trigonometry
SWBAT:
- Understand congruence of triangles and apply properties for problem solving.
- Understand similarity, right triangles and trigonometry.
- Understand and apply the properties of quadrilaterals to solve problems.
- Understand and perform transformations.
Unit 4: Circles
SWBAT:
- Understand and apply properties of circles to solve problems.
Unit 5: Three-Dimensional Geometry
SWBAT:
- Understand and calculate surface area and volume of figures.
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Pacing Guide- Course
Content
Number of Days
Unit 1:
Geometric Representations
20
Unit 2:
Reasoning in Geometric Situations
20
Unit 3:
Congruence, Similarity and Right Triangle Trigonometry
90
Unit 4:
Circles
30
Unit 5:
Three-Dimensional Geometry
20
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Curriculum Units
Unit 1: Geometry Representations:
Essential Questions
Instructional Objectives/
Skills and Benchmarks
Activities
Assessments

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Identify objects in the classroom that can be
represented by geometric shapes.
 Calculate distances on the coordinate plane.

Find distances and midpoints.

Students, in their study of algebra, learned
ways to find the point of intersection of two
lines. They should connect this to problems
associated with specific geometric
representations (e.g., finding the point at
which a perpendicular bisector intersects a
line segment on the coordinate plane).
Enrichment: Study coordinates in space as
an extension of 3D geometry..
How can geometric shapes be
used to represent the real world.

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How can coordinates be used to
describe and analyze geometric
objects?

Model points, lines and planes.
(G-CO.1)
Use segments, midpoints,
distance formula and angles to
solve problems
(G-CO.1)
Use coordinates and algebraic
techniques to describe and define
geometric figures and to
interpret, represent, and verify
geometric relationships.
(G-GPE.4 – 7)
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How are vectors used?
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Show position and represent
motion in the coordinate plane
using vectors
(G-CO.2)
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Students will perform transformations using
vectors.
 Extended Constructed Response (ECR): Given
the coordinates of the vertices of quadrilateral,
determine whether it is a parallelogram.
 SCR: The rectangular coordinates of three
points in a plane are Q(-3, -1), R(-2, 3), and S(1,
-3). A fourth point T is chosen so that the
magnitude of vector ST is equal to twice the
magnitude of vector QR. What is the ycoordinate of T?

Calculate positions using vector notation.
Unit 2: Reasoning in Geometric Situations:
Essential Questions
Instructional Objectives/
Skills and Benchmarks
Activities
Assessments
*How can mathematical
reasoning help you make
generalizations?
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Complete a proof for an algebraic solution.
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Prove the same statement (e.g., the
diagonals of a rectangle are congruent)
using a proof.
ECR: Write the inverse, converse, and
contrapositive of an advertising slogan and
identify which of these statements are logically
equivalent.
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Technology Integration: Use
Classzone.com to analyze angle
relationships.
ECR: Prove that the diagonals of a rhombus
are perpendicular. (This might be done using
Euclidean methods or coordinate geometry).

ECR: Given a line and a point, construct a line
through the point that is perpendicular to the
original line. Justify the steps of the
construction. (Note that students might use
tools of their choice to answer this question.)
MC: The vertices of a triangle PQR are the
points P(1, 2), Q(4, 6), and R(-4, 12). Which
one of the following statements about triangle
PQR must be true?
*How do you know when
you have proven
something?
*How can
geometric/algebraic
relationships best be
represented and verified?
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Use reasoning and some form of proof to
verify or refute conjectures and theorems
about lines, angles and polygons.
(G-CO.9)
Students should be able to:
 Make, test, and confirm or refute geometric
conjectures using a variety of methods,
including technology.
 Demonstrate through example or explanation
how indirect reasoning can be used to
establish a claim.
 Recognize flaws or gaps in the reasoning
supporting an argument.
 Develop a counterexample to refute an
invalid statement.
 Determine the correct conclusions based on
interpreting a theorem in which necessary or
sufficient conditions in the theorem or
hypothesis are satisfied.
(G-GPE.4 – 7)
Basic concepts include the following:
 Vertical Angles
 Supplementary and complementary angles
 Linear pairs of angles
 Angles formed by parallel lines (alternate
interior and corresponding angles)
 Perpendicular bisector
 Angle bisector
 Midpoint and perpendicular bisector of a
segment
 Perpendicular or parallel lines
(G-CO.9-11)
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A.
PQR is a right triangle with the right angle
at P.
B. PQR is a right triangle with the right angle
at Q.
C. PQR is a right triangle with the right angle
at R.
D. PQR is not a right triangle.
Unit 3: Congruence, Similarity, and Right Triangle Trigonometry
Essential Questions
Instructional Objectives/
Skills and Benchmarks
Activities
Assessments
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What makes shapes
alike and different?
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Determine and apply conditions that
guarantee congruence of triangles.
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How are similarity,
congruence, and
symmetry related?
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Analyze special segments within triangles.
ECR: Use congruent triangles to prove that the
bisector of the angle opposite the base of an
isosceles triangle is the perpendicular bisector
of the base.
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Identify and apply conditions that are
sufficient to guarantee similarity of
triangles.
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Find the scale factor for two congruent
polygons.
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What situations can be
analyzed using
transformations and
symmetries?
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Extend the concepts of similarity and
congruence to other polygons in the plane.
How can
transformations be
described
mathematically?
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Show how similarity of right triangles
allows the trigonometric functions sine,
cosine, and tangent to be properly defined
as ratios of sides.
How can we describe
and analyze iterative
geometric patterns?
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Analyze properties of parallelograms and
other quadrilaterals.
(G-C0.6-8, G-SRT.1-11)
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Interdisciplinary Connections: Use
similarity to calculate the measures of
corresponding parts of similar figures, and
apply similarity in a variety of problem
solving contexts within mathematics and
other disciplines.
Classroom Task: Have each student draw
their own triangle and then construct a line
parallel to one side. Ask them how the
smaller triangle is related to the larger one.
Have them explain and justify their
conjectures.
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Design the perfect bedroom and make a
scale drawing.
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Enrichment: Derive, interpret, and use the
identity sin2θ + cos2θ = 1 for angles θ
between 0° and 90° as a special
representation of the Pythagorean
Theorem.
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Performance Assessment Task: Measure the
heights of a number of objects and the lengths
of their shadows on a sunny day. Create a
scatterplot of the data, and write an equation
that relates shadow length to object height.
Write a brief report of your work and your
findings, describing how you used similar
triangles. (NCTM Navigating Through
Geometry)
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SCR: The maximum slope for handicapped
ramps in new construction is 1/12. What
angle will such a ramp make with the ground?
Unit 4: Circles
Essential
Questions
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What geometric
relationships can
be found in circles
Instructional Objectives/ Skills and
Benchmarks
Activities
Assessments
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To investigate the intersecting
chords theorem and other
segment relationships using the
computer
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Prove that if a radius if a circle is
perpendicular to a chord of the
circle, then it bisects the chord
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Find approximate area of a
running track consisting of a
rectangle and two semi-circles
Show that a triangle inscribed on
the diameter of a circle is a right
triangle
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Technology Integration: Use a
graphing calculator to graph
equations of circles
An apple pie is cut into six equal
slices. The diameter of the pie is
10 inches. Find the approximate
arc length of one slice of pie and
the approximate area of the top of
one slice of pie
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Find the equation of a circle with
diameter 20in and center at the
(3,-1)
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To recognize and apply the definitions and
properties of a circle
To verify and apply relationships associated
with a circle
To determine the length of line segments and
arcs, the measure of angles, and the areas of
shapes that they define in complex geometric
drawings
To relate the equation of a circle to its
characteristics and its graph
(G-C.1-4)
Unit 5: Three-Dimensional Geometry
Essential Questions
Instructional Objectives/ Skills
and Benchmarks
Activities
Assessments
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How are 1-, 2-, and 3dimensional shapes related?
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How can 3-dimensional objects
be represented in 2 dimensions?
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How can 3D objects be
measured?
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To determine the surface area and
volume of solid figures,
recognizing and using
relationships among volumes of
common solids
To create, interpret, and use twodimensional representations of 3dimensional shapes
To analyze cross-sections of
basic three-dimensional objects
and identify the resulting shapes
To describe the characteristics of
3-dimensional objects traced out
when a one- or two- dimensional
figure is rotated about an axis
To analyze all possible
relationships among two or three
planes in space and identify their
intersections
(enrichment topics not included
in CCCS)
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Read Flatland and discuss how
1D, 2D, 3D and 4D world are
related.
Use appropriate formula to
calculate 3D measures.
Use a clear transparency to
construct a net of a 3-D figure
and then make the object
Use different shaped boxes to
show how they are different or
the same from the mathematical
planes that they represent.
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Complete a project based on
Flatland to evaluate the themes of
the book.
Find the surface area of a square
pyramid whose sides all have
length 2 cm.
Find the volume of a cone with
radius 2in, and height 4in
Sketch the figure formed of
blocks from perspective views
using isometric dot paper.
Describe all possible results of
the intersection of a plane with a
cube
A triangle has vertices at (0,0),
(2,0), and (0,10). What is the
volume of the shape generated
when this triangle is rotated about
the x-axis?
New Jersey Common Core State Standards
Geometry
Congruence
Experiment with transformations in the plane

G.CO.1. Know precise definitions 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.
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G-CO.2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in
the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus
horizontal stretch).
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G-CO.3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
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G-CO.4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
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G-CO.5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry
software. Specify a sequence of transformations that will carry a given figure onto another.
Understand congruence in terms of rigid motions
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G-CO.6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two
figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
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G-CO.7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and
corresponding pairs of angles are congruent.
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G-CO.8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
Prove geometric theorems
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G-CO.9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate
interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant
from the segment’s endpoints.
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G-CO.10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are
congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

G-CO.11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a
parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
Make geometric constructions
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G-CO.12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding,
dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines,
including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
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G-CO.13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Similarity, right triangles and Trigonometry
Understand similarity in terms of similarity transformations

G-SRT.1. Verify experimentally the properties of dilations given by a center and a scale factor:

A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
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The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
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G-SRT.2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity
transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs
of sides.
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G-SRT.3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
Prove theorems involving similarity
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G-SRT.4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely;
the Pythagorean Theorem proved using triangle similarity.
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G-SRT.5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
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Define trigonometric ratios and solve problems involving right triangles

G-SRT.6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios
for acute angles.
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G-SRT.7. Explain and use the relationship between the sine and cosine of complementary angles.
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G-SRT.8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. ★
Apply trigonometry to general triangles
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G-SRT.9. (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
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G-SRT.10. (+) Prove the Laws of Sines and Cosines and use them to solve problems.
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G-SRT.11. (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g.,
surveying problems, resultant forces).
Circles
Understand and apply theorems about circles
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G-C.1. Prove that all circles are similar.
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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.
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G-C.3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
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G-C.4. (+) Construct a tangent line from a point outside a given circle to the circle.
Find arc lengths and areas of sectors of circles
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G-C.5. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the
angle as the constant of proportionality; derive the formula for the area of a sector.
Expressing Geometric Properties with equations
Translate between the geometric description and the equation for a conic section

G-GPE.1. 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.
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G-GPE.2. Derive the equation of a parabola given a focus and directrix.
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G-GPE.3. (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.
Use coordinates to prove simple geometric theorems algebraically

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).
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G-GPE.5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or
perpendicular to a given line that passes through a given point).
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G-GPE.6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
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G-GPE.7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.
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