Download Goemetry Gallery and Coordinate Geometry

THE MATH BUZZ: HIGH SCHOOL MATHEMATICS 1
A Mathematics Resource for Georgia Cyber Academy Parents and Students
GEOMETRY GALLERY &
COORDINATE GEOMETRY
KEY
CONCEPTS
For the next few months we will cover the two main Geometry units in Georgia’s Math 1
curriculum entitled Geometry Gallery (Unit 4) and Coordinate Geometry (Unit 5). In your
LMS we will start in MTH123B Algebra 1 Unit 7 then move to MTH203A Geometry Units
1, 3, 4 and 5.
GEORGIA PERFORMANCE
STANDARDS
GEOMETRY
MM1G1. Students will investigate properties of
geometric figures in the coordinate plane.
 MM1G1.a. Determine the distance between two
points.
 MM1G1.b. Determine the distance between a
point and a line.
 MM1G1.c. Determine the midpoint of a segment.
 MM1G1.d. Understand the distance formula as
an application of the Pythagorean theorem.
 MM1G1.e. Use the coordinate plane to
investigate properties of and verify conjecture
related to triangles and quadrilaterals.
MM1G2. Students will understand and use the
language of mathematical argument and
justification.
 MM1G2.a. Use conjecture, inductive reasoning,
deductive reasoning, counterexamples, and
indirect proof as appropriate.
 MM1G2.b. Understand and use the relationships
among a statement and its converse, inverse,
and contrapositive.
MM1G3. Students will discover, prove, and
apply properties of triangles, quadrilaterals,
and other polygons.
 MM1G3.a. Determine the sum of interior and
exterior angles in a polygon.
 MM1G3.b. Understand and use the triangle
inequality, the side-angle inequality, and the
exterior-angle inequality.
 MM1G3.c. Understand and use congruence
postulates and theorems for triangles (SSS, SAS,
ASA, AAS, HL).
 MM1G3.d. Understand, use, and prove
properties of and relationships among special
quadrilaterals: parallelogram, rectangle,
rhombus, square, trapezoid, and kite.
 MM1G3.e. Find and use points of concurrency in
triangles: incenter, orthocenter, circumcenter,
and centroid.
Have you ever had an argument with your parents? Did you win? In these units we will
show you how to construct a valid argument. You will explore, understand and use the
formal language of reasoning and justification which could help you in your next
disagreement. You will learn how to apply this logical reasoning to proofs in geometry.
You will also discover, prove and apply the properties of triangles, quadrilaterals and
other polygons. Finally you will combine all that you have learned regarding algebra and
geometry and explore geometric shapes on the coordinate plane.
The two main essential questions that we will strive to answer in these units are as
follows:
 What are the properties of convex polygons and how are they proven?
 How do algebra and geometry work together within the coordinate plane?
Please remember to follow your Course Calendar Announcements found in the
announcement section of the LMS very closely, complete all assessments on time and
attend your ClassConnect Sessions. These are your keys to success. If at any time you
feel lost or confused please contact your math teacher or anyone in our math department.
We are here to help.
9th Grade Math Department
Helpful Math Websites
Visual Dictionary Visual Mathematics Dictionary
Video Library http://algebasics.com/index.html
Graphing Calculator
http://my.hrw.com/math06_07/nsmedia/tools/Graph_Calculator/graphCalc.html
Geometry - http://www.mathopenref.com/
All Subjects - http://www.onlinemathlearning.com/
Logic Statements
http://www.sparknotes.com/math/geometry3/logicstatements/section3.rhtml
Conditional Statement
http://www.explorelearning.com/index.cfm?method=cResource.dspDetail&ResourceID=1
78
Hypothesis/Conclusion http://www.slideshare.net/rfant/hypothesis-conclusion-geometry14
Triangle Inequality http://www.onlinemathlearning.com/triangle-inequality.html
Congruent Triangles http://www.mathopenref.com/congruenttriangles.html
Exterior & Interior Angles of a Polygon
http://www.mathopenref.com/polygonanglerelation.html
Triangle Centers http://www.mathopenref.com/trianglecenters.html
Quadrilaterals http://www.mathopenref.com/keywordindices/quadrilateral.html
Midpoints http://www.mathsnet.net/dynamic/cindy/exploration_2.html
UNIT 4 GEOMETRY GALLERY & UNIT 5 COORDINATE GEOMETRY
VOL 1 ISSUE 4
KEY VOCABULARY
1.
2.
3.
4.
Conjecture: an educated guess or a theorem that is yet to be proven
Contrapositive: a conditional statement that both switches and negates the hypothesis and the conclusion of the original conditional statement
Converse: the statement that is formed when the "if" and "then" of an original statement are interchanged
Counterexample: an example that shows a statement is not always true
5.
6.
7.
Deductive reasoning: reasoning from the general to the specific
Inductive reasoning: reasoning from the specific to the general
Inverse: conditional statement that negates both the hypothesis and the conclusion of the original conditional statement
8.
9.
Proof: logical reasoning that uses facts, definitions, properties, and previously proven theorems to show that a proposition is true
Syllogism (SIH-luh-jih-zuhm): a logical argument that always contains two premises and a conclusion; syllogisms have the following form: If a,
then b. If b, then c. Therefore, if a, then c. It is also known as the Law of Syllogism.
Theorem (THIR-uhm) a statement that is shown to be true by use of a logically developed argument
Points of Concurrency: The point where three or more lines intersect. (Usually refers to various centers of a triangle)
Centroid: The point of concurrency of the medians of a triangle.
Medians of a Triangle: A median of a triangle is a line joining a vertex to the midpoint of the opposite side.
Circumcenter: The point of concurrency of the perpendicular bisectors of the sides of a triangle.
Perpendicular Bisectors: A line which cuts a line segment into two equal parts at a 90° angle.
Incenter: The point of concurrency of the bisectors of the angles of a triangle.
Angle Bisector: A line which cuts an angle into two equal halves
Orthocenter: The point of concurrency of the altitudes of a triangle.
Altitudes of a Triangle: a straight line through a vertex and perpendicular to (i.e. forming a right angle with) the opposite side or an extension of
the opposite side.
Sum of the measures of the interior angles of a convex polygon: 180º(n – 2).
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21. Measure of each interior angle of a regular n-gon:
22.
23.
24.
25.
180  (n  2)
n
Exterior angle of a polygon: an angle that forms a linear pair with one of the angles of the polygon.
Remote interior angles of a triangle: the two angles non-adjacent to the exterior angle.
Measure of the exterior angle of a triangle: equals the sum of the measures of the two remote interior angles.
Exterior Angle Inequality: an exterior angle is greater than either of the remote interior angles.
26. Distance Formula: d =
( x2  x1 ) 2  ( y2  y1 ) 2
 x1  x2 y1  y 2 
,


2
2 
27. Midpoint Formula: 
28. SSS Theorem: If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.
29. SAS Theorem: If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the
triangles are congruent.
30. ASA Theorem: If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the
triangles are congruent.
31. AAS Corollary: ASA leads to the AAS corollary. If two angles on a triangle are known, then the third is also known.
32. HL Theorem: If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle,
then the triangles are congruent.
UNIT 4 GEOMETRY GALLERY & UNIT 5 COORDINATE GEOMETRY
VOL 1 ISSUE 4