Find the coordinates of the midpoint of a segment having the given endpoints.

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Martha Ramsey
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1 G.(2) Coordinate and transformational geometry. The student uses the process skills to understand the connections between algebra and geometry and uses the one- and two-dimensional coordinate systems to verify geometric conjectures. The student is expected to: G.2(A) determine the coordinates of a point that is a given fractional distance less than one How can we find the midpoint of a line segment on a number line or in a coordinate plane? Find the coordinates of the midpoint of a segment having the given endpoints. Coordinate Distance Formula Midpoint Formula Compare the methods of counting lines on the number line or coordinate plane and using the from one end of a line SAT: Find the coordinates of midpoint or distance 3.5 segment to the other the other endpoint given one A. (-1,8) formula to calculate the 3.6 in one- and twodimensional coordinate midpoint of (6, 1) C. (6,1) endpoint of (1, -3) and a B. (-10,-8) distances. systems, including A. (-1, 8) D. (-5,-4) finding the midpoint. B. (-5, -4) C. (11, 5) D. (10, 8) G2(B) derive and use the distance, slope, and midpoint formulas to verify geometric relationships, including congruence of segments and parallelism or perpendicularity of pairs of lines. Compare the Distance Formula to the Pythagorean Theorem Given two points on the coordinate grid, (1, 5) and (2,-2), find the distance between the points in reduced radical form. Answer:5 2 Graph the points D (3, 4) E (0, 4) and F (-7, 4). Connect the points with a segment. Calculate mde and EF. Use the Segment Addition Postulate (Postulate 1.2) to find the length of DF. Verify the length of DF using the Distance Formula. Demonstrate how the Pythagorean Theorem can be used to calculate distances and how the Distance formula is related to the Theorem Page 1

2 G.(4) Logical argument and constructions. The student uses the process skills with deductive reasoning to understand geometric relationships. The student is expected to: G.4(A) distinguish between undefined terms, definitions, postulates, conjectures, and theorems. Develop verbal descriptions to define Geometric Terms throughout the curriculum Use number line and coordinate plane to represent points, lines, rays, line segments and geometric figures. If two lines intersect, then their intersection is exactly one point. Is this a definition, a postulate or an undefined term? Answer: Postulate Conjecture Definition Postulate Theorem Undefined Term Develop verbal descriptions to define geometric terms throughout the curriculum Use manipulatives and technology to draw conclusions and discover relationships about geometric shapes and their G.4(B) identify and determine the validity of the converse, inverse, and contrapositive of a conditional statement and recognize the connection between a biconditional statement and a true conditional statement with a true converse. Make, interpret and/or understand statements such as if p, then q as applied to attributes of geometric drawings, figures, etc. Develop conjectures in the form of a conditional statement Use counter-examples to prove why statements are false Use inductive or deductive reasoning to prove statements true. Given a conditional statement: If you are a guitar player, then you are a musician. Write the converse, inverse and contrapositive of the statement and state whether each statement is true or false. Can the statement and converse be a biconditional? If not, give a counterexample. Conclusion Conditional Statement Contra-Positive Converse Hypothesis Inverse Negation Truth Value properties Write conditional statements, converse, inverse and contrapositive Use discussions and brainstorming to determine the validity of each statement and provide a counter-example, if false Page 2

3 G.4(C) verify that a conjecture is false using a counterexample. Draw conclusions from number or picture patterns, specific examples or events. Draw conclusions by using inductive reasoning to form conjectures Determine if the following conjecture is true or false: The value of x^2 is always greater than the value of x. If false, give a counter-example. Conjecture Counter-example Use facts, definitions, postulates, theorems and properties to prove statements true or false. 2.2 G.(5) Logical argument and constructions. The student uses constructions to validate conjectures about geometric figures. The student is expected to: G.5(A) investigate patterns to make conjectures about geometric relationships, including angles formed by parallel lines cut by a Draw conclusions from number or picture patterns, specific examples or events. Draw conclusions by using inductive reasoning to form How are angles 1 and 8 related? Alternate Exterior Alternate Interior Corresponding Teacher may wish to use patty paper for students to draw/construct parallel lines and investigate angle pairs transversal, criteria conjectures A. Same side interior required for triangle B. Alternate exterior Diagonal congruence, special C. Alternate interior Parallel Lines segments of triangles, D. Corresponding Perpendicular diagonals of Lines quadrilaterals, interior Answer: B Same-side Interior and exterior angles of polygons, and special segments and angles of circles choosing from a Segment Skew Lines Transversal variety of tools Patty Paper Geometry by Michael Serra Page 3

G.5(B) construct congruent segments, congruent angles, a segment bisector, an angle bisector, perpendicular lines, the perpendicular bisector of a line segment, and a line parallel to a given line

4 G.5(B) construct congruent segments, congruent angles, a segment bisector, an angle bisector, perpendicular lines, the perpendicular bisector of a line segment, and a line parallel to a given line through a point not on a line using a compass and a straightedge. Compare and contrast sketches, drawings and constructions as graphic representations of geometric figures. What construction is shown in the accompanying diagram? A. The bisector of angle PJR. B. The midpoint of line PQ C. The Perpendicular bisector of line segment PQ. D. A perpendicular line to PQ through point J. Compass Construction Drawing Sketch Straight Edge Teacher can demonstrate construction techniques using in-class document camera or go to the math open reference website for animations demonstrating constructions. f.com/tocs/construction stoc.html Page 4

5 G.(6) Proof and congruence. The student uses the process skills with deductive reasoning to prove and apply theorems by using a variety of methods such as coordinate, transformational, and axiomatic and formats such as two-column, paragraph, and flow chart. The student is expected to: G.6(A) verify theorems about angles formed by How are the different angle pairs related? Adjacent Alternate Exterior Explore different proof methods to verify the 1.5 the intersection of lines theorems concerning 1.6 and line segments, How do their measures Based on the illustration, what is the Alternate Interior segments, angles and their including vertical angles, and angles formed by parallel lines cut by a transversal and prove equidistance between the endpoints of a segment and points compare? How and why do we know this? Corresponding Diagonal Linear Pairs Parallel Lines relationships. on its perpendicular Perpendicular bisector and apply these relationship between angle 1 and Lines relationships to solve angle 3? Same-side Interior problems. A. They are congruent angles. B. They are supplementary. Segment C. They form a right angle. Skew Lines Supplementary D. They are complementary. Answer: A Transversal Page 5

Geometry Vocabulary Book

Geometry Vocabulary Book Units 2-4 Page 1 Unit 2 General Geometry Point Characteristics: Line Characteristics: Plane Characteristics: RELATED POSTULATES: Through any two points there exists exactly one