# Coordinate Coplanar Distance Formula Midpoint Formula

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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 oneand two-dimensional coordinate systems to verify geometric conjectures. G.2(A) determine the coordinates of a point that is a given fractional distance less than one from one end of a line segment to the other in one- and twodimensional coordinate systems, including finding the midpoint. Determine the coordinates of a point that is given fractional distance from either endpoint of a line segment. Find the coordinates of point P along the directed line segment AB so that AP to PB is the ratio 2 to 6 when A(-3,2), B(5,-4). Correct answer: P(-1,0.5) Coordinate Coplanar Distance Formula Midpoint Formula Compare the methods of counting lines on the number line or coordinate plane and using the midpoint or distance formula to calculate the distances. Big Ideas 3.5, Example 2 Connects to G.2B Page 1

2 G.2(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. Readiness Standard Use slopes to determine whether the lines are parallel, perpendicular or neither. Comparing equations of lines, determine whether lines are parallel, intersect or coincide. Prove lines are parallel given angle information. Prove and apply theorems about perpendicular lines State whether the graphs of the following equations are parallel, perpendicular, intersecting or coincide Ex. 5x-3y = 7 y= -3/5x + 8 Correct answer: Perpendicular Midpoint Slope Parallel Perpendicular Interesting lines Coincide Slope intercept form y- intercept Use distance formula to find the distance between 2 points. Find coordinates of the midpoint of a segment on a coordinate plane. Find endpoint given an endpoint and a midpoint. Demonstrate through use and problem solving. 3.4, 3.5, 3.6, 5.1 Misconceptions: The student may substitute the x- and y-values incorrectly when using the formulas. The student may divide a value by 2 instead of taking the square root when using the distance formula. The student may add the x-value to the y-value, instead of computing the sum of the x-values and computing the sum of the y-values before dividing by 2 in the midpoint formula. The student may incorrectly write the ratio of the slope of a line as the ratio of horizontal change divided by vertical change Page 2

3 G.2(C) determine an equation of a line parallel or perpendicular to a given line that passes through a given point. Readiness Standard Write and compare equations of lines. Write the equation of a line parallel and perpendicular to a given line through the given point. The graph of line g is shown below. What equation describes a line parallel to g that has a y-intercept at (0,-1)? Correct answer: 1 y x 1 Released EOC 2013 Q#25 2 Slope intercept form y- intercept perpendicular- bisector coordinate Review the relationships between slopes of parallel and perpendicular lines. Provide students with graphic organizers to help sort parallel and perpendicular lines. 3.5, 3.6 Google Drive: G.2C Task Activity Use Guided Practice G.2C Task Activity in the Google drive for class practice. Misconceptions: The student may use the slope formula incorrectly horizontalchange (ie: instead of vertical change ). verticalchange horizontal change The student may think the slopes of perpendicular line are only opposite values instead of opposite reciprocals Page 3

4 G.(3) Coordinate and transformational geometry. The student uses the process skills to generate and describe rigid transformations (translation, reflection, and rotation) and non-rigid transformations (dilations that preserve similarity and reductions and enlargements that do not preserve similarity). G.3(A) describe and perform transformations of figures in a plane using coordinate notation. Describe transformations of figures in a plane using coordinate notation. Parallelogram ABCD was transformed to form parallelogram A B C D. Transformation Translation Reflection Rotation Big Ideas 4.1 Dilation Connects to G.3B Perform transformations of figures in a plane using coordinate notation. Which rule describes the transformation that was used to form parallelogram A B C D? F. ( x, y) ( x, y) G. ( x, y) ( x, y) H. ( x, y) ( x 6, y) J. ( x, y) ( x, y 3) (-x, y-3) Correct answer: J Adapted from Released EOC 2013 Q#40 Use three column notes to provide transformation in a plane in one column, verbal description in another, coordinate notation in third. Google Drive: Graphic Organizer Card Sort Activity Engaging p. 45 (18.pdf) Page 4

5 G.3(B) determine the image or pre-image of a given two-dimensional figure under a composition of rigid transformations, a composition of non-rigid transformations, and a composition of both, including dilations where the center can be any point in the plane. Readiness Standard Determine the image pre-image of a given twodimensional figure under a composition of rigid non-rigid both transformations. Determine the image pre-image of a given twodimensional figure that includes dilations where the center can be any point in the plane. ΔABC has vertices A(-3, 1), B (2, - 1), and C (0, 2). Reflect the figure across the y-axis and then translate it 3 units down and 4 units to the right. What are the coordinates of the image? Correct Answer: A (7, -2), B (2, -4), C (4, -1) Image Pre-image Transformation Translation Reflection Rotation Dilation Composition Center of Dilation Rigid transformation Non/Congruent figures Center of dilation at origin: Multiply coordinates of preimage by scale factor Center of dilation not at origin: use slope to find image points Stress use of prime notation for image points Big Ideas 4.1, 4.2, 4.6 Engaging p. 57 (23.pdf) & p. 59 (24.pdf) Misconceptions: The student may not be able to distinguish the difference between image and pre-image The student may think the origin is the only point that can be the center for dilations Page 5

6 G.3(C) identify the sequence of transformations that will carry a given pre-image onto an image on and off the coordinate plane. Connects to G.3B Identify the sequence of transformations that will carry a given pre-image onto an image on the coordinate plane. Identify the sequence of transformations that will carry a given pre-image onto an image off the coordinate plane. Jake took pictures of Ana s flag while she was practicing her routine for the football game, as shown below. Which of the following best describes the movement of the flag from picture to picture? A. Reflection, rotation, translation B. Rotation, translation, translation C. Rotation, translation, dilation D. Reflection, translation, translation Image Pre-image Transformation Translation Reflection Rotation Dilation Composition Center of Dilation Point of rotation Scale factor Similarity Demonstrate that the order of the transformations matters. Include a variety of examples where students identify the sequence of transformations. There may be several different methods for transforming the same pre-image into an image. Big Ideas 4.4, 4.6 G.3(D) identify and distinguish between reflectional and rotational symmetry in a plane figure. Connect to G.3B Identify reflectional symmetry in a plane figure. Identify rotational symmetry in a plane figure. Distinguish between reflectional and rotational symmetry in a plane figure. Answer: A Tell whether the figure has rotational and/or reflectional symmetry. Rotational yes Reflectional no Symmetry Rotational symmetry Reflectional symmetry Line of reflection Line of symmetry Center of rotation Angle of rotation Center of symmetry Reflectional: over a line Rotational: about a point Make sure students label the vertices Big Ideas 4.2, Page 6

7 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 transversal, criteria required for triangle congruence, special segments of triangles, diagonals of quadrilaterals, interior and exterior angles of polygons, and special segments and angles of circles choosing from a variety of tools. Readiness Standard Use the exterior angle theorem to find angle measures Find the measure of F Exterior angles theorem Exterior angle Interior angle Remote interior angles Triangle sum theorem Use both numeric and algebraic expressions to find missing angles measures 5.2 Misconceptions: The student may make a conjecture based on limited investigation of patterns. The student may randomly state a conjecture without investigating and recognizing patterns. The student may not know how to use a construction to make a conjecture. The student may not be able to perform constructions correctly. The student may not state a conjecture using precise geometric vocabulary. Engaging p. 79 (32.pdf) Page 7

8 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. Connects to G.5A, G.6A G.5(C) use the constructions of congruent segments, congruent angles, angle bisectors, and perpendicular bisectors to make conjectures about geometric relationships. Connects: G.5A, G.6A Use a compass and a straight edge to construct: Perpendicular lines The perpendicular bisector of a line segment using a compass and a straightedge. A line parallel to a given line through a point not on a line Use constructions, Congruent segments Congruent angles Angle bisectors Perpendicular bisectors To make conjectures about geometric relationships. 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. Answer: C Compass Construction Drawing Sketch Straight Edge Angle bisector Bisect Congruent Congruent angles Congruent segments Constructions Perpendicular Perpendicular bisector Focus on constructing geometric figures with only a straight edge and a compass. Ensure students can construct congruent segments. Use two column notes that have students write the steps needed to construct on one side while performing the task of construction in the other. As students construct figures, they should also describe what they see and explain why the construction works. Big Ideas 3.3, 3.4 com/tocs/constructionsto c.html 3.3, 3.4, Page 8

9 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 the intersection of lines Find the value of x and solve problems involving parallel lines and vertical Find the value of x to verify that the lines are parallel Alternate Exterior Angles Alternate Interior Substitute different values to verify angle measures. Big Ideas 3.3, 3.4 and line segments, angles Angles Use manipulatives and including vertical Coplanar technology to draw angles, and angles Corresponding Angles conclusions and formed by parallel lines Diagonal discover relationships cut by a transversal and Graph segments and find the Parallel Lines about parallel lines and prove equidistance perpendicular bisector using PQ is shown on the coordinate Perpendicular Lines their properties between the endpoints the slope and midpoint Same-Side Interior of a segment and points formulas grid below. The coordinates of P Angles Stress the importance of on its perpendicular and Q are integers. Segment slopes perpendicular to bisector and apply these Skew Lines a line (opposite relationships to solve Transversal reciprocal) problems. Readiness Standard Point (x, y) lies on the perpendicular bisector of PQ. What is the value of x? Correct answer: -2.5 Released EOC 2013 Q#10 bisector Slope Midpoint Coordinates Use distance formula to find the distance between 2 points and the midpoint Find coordinates of the midpoint of a segment on a coordinate plane. Misconceptions: The student may not use logical reasoning correctly to work through proofs. The student may not apply justification to support statements in a twocolumn proof Page 9

10 G.6(C) apply the definition of congruence, in terms of rigid transformations, to identify congruent figures and their corresponding sides and angles. Connects to G.6B, G.3B G.6(D) verify theorems about the relationships in triangles, including proof of the Pythagorean Theorem, the sum of interior angles, base angles of isosceles triangles, midsegments, and medians, and apply these relationships to solve problems. Connects to G.5A Apply the definition of congruence, in terms of rigid transformations, to identify congruent figures corresponding sides (of congruent figures) corresponding angles (of congruent figures) Verify theorems about the relationships in triangles: Including the sum of interior angles. (The rest of this SE is addressed in the 3 rd and 4 th grading periods.) Find the missing angle measure in triangles Determine relationships of angles and sides when bisectors, medians and altitudes are drawn in triangles. Use AAS to explain why the triangles are congruent. Answer: A D, BEA CED, BE CE B is the midpoint of A B D D is the midpoint of and AE = 21. Find BD. The diagram is not to scale. C E Corollary Corresponding Angles Corresponding Polygons Corresponding Sides Included Angle Included Side Interior Triangle Rigidity SAS SSS ASA AAS AL Midsegment Midpoint Congruent Parallel Isosceles triangle equilateral triangle Equidistant Base angles Medians, bisectors Hinge theorem Inequality Perpendicular bisector Altitude Students should mark pictures with congruence to be able to easily determine how the triangles are congruent i.e. AAS, SAS, ASA Verify relationships in triangles including triangle sum theorem, base angles of isosceles triangles and angles in equilateral triangles. In an Isosceles triangle, have students discover Median, angle bisector, perpendicular bisector are all the same line. Find the value of the midsegment given the parallel side of the triangle. Use both algebraic expressions and numeric values when solving 4.4, , 5.4 Engaging p (39.pdf) Page 10

11 G.7 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.7(A) apply the definition of similarity in terms of a dilation to identify similar figures and their proportional sides. Apply the definition of similarity in terms of a dilation to identify similar figures. Isosceles trapezoid JKLM is shown below. Congruent corresponding angles Dilation Proportional Similar figures Similarity Utilize a graphic organizer to compare the properties of congruence transformation and similarity 4.6 transformations. Connects to G.3B, G.7B (The rest of this SE is addressed in the 4 th grading period.) If the dimensions of the trapezoid JKLM are multiplied by a scale factor of f to create trapezoid J K L M, which statement is true? F. Trapezoid J K L M contains two base angles measuring 30 each. G. The longer base of trapezoid J K L M is 56f units. H. The bases of trapezoid J K L M have lengths of 22 units and 39 units. J. Trapezoid J K L M contains two base angles measuring (120 f ) each. Correct answer: G Released EOC 2013 Q# Page 11

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