GSMST Class of 2018



Isometry- does not change shape or size of figure, only changes location or positionTypes of transformations:ReflectionRotationTranslationDilationA dilation is not an isometry.A transformation is written PREIMAGE IMAGEThe preimage is the original figure. The image is the translated figure, and is written with primes.Rotations are always counterclockwise unless specified otherwiseA composition of transformations is one transformation followed by others.Reflection across y axis(x,y) (-x,y)Reflection across x axis(x,y) (x, -y)Rotation (90 degree)(x,y) (y, -x)Rotation (180 degree)(x,y) (-x, -y)Rotation (270 degree)(x,y) (-y, x)Translation(x, y) (x +a, y + b)A and B are negative/ subtraction for left or down and positive for right or up.Dilation(x,y) (ax, by)Each is multiplied by a scale factor. Fractional scale factor/dividing means getting smaller.CONSTRUCTING TRANSLATIONS NOTESA vector has distance and directionWhen using a vector to translate a figure, whatever part of the figure is at the tail of the vector will be moved to the tip of the vector.Vectors MUST be written with brackets. They’re not a point, they show change.Example: <1,2> Shows a movement of one right and two up.Flip the signs of the vector to go from the image back to the preimageREFLECTIONS NOTESReflections can be used to give you the shortest distance between two points.Think of what values are staying the same- x or y- to determine what line you are reflecting over. TheoremThe composition of two reflections across two parallel lines is equivalent to a translation in that same direction. The translation vector is perpendicular to the two linesThe length of the translation vector is twice that of the distance between the two linesTheorem-The composition of two reflections across intersecting lines is equivalent to a rotation.The center of rotation is the intersectionThe angle of rotation is twice the angle of intersectionTRANSLATIONS NOTESGlide translation-A glide translation is a translation followed by a reflection.Frieze pattern-This is a zig zag pattern.Tessellation-A tessellation has angles that add up to 360 degrees in each place that they meet.SYMMETRY NOTESA figure has symmetry if there is a transformation where the image coincides with the preimage.A LINE OF SYMMETRY divides a figure into two congruent halves.LINE SYMMETRY is when a figure can be reflected over a line of symmetry to coincide with itself.ROTATIONAL SYMMETRY is when a figure can be rotated about a point by an angle less than 360 and greater than 0 degrees, so that it coincides with itselfThe ANGLE OF ROTATIONAL SYMMETRY is the smallest angle you can rotate a figure by for it to coincide with itselfThe ORDER is the number of times a figure can be rotated to coincide with itself. ORDER is 360/angle of rotational symmetryPLANE SYMMETRY is when you can divide a 3D figure with a plane, and both portions of it can be reflected across that plane to coincide.SYMMETRY ABOUT AXIS is when a figure can be rotated about an axis and coincide with itself at some point less than a 360 degree rotation.TRANSLATIONAL SYMMETRY is when a pattern can be translated along a vector and one part coincides with another.PERPENDICULAR LINES NOTESTwo lines that are perpendicular meet at a 90 degree angleTheir slopes cancel out to equal one. For examplem= 3/4 and m= -4/3 are perpendicularMIDPOINTmiddle pointthe midpoint formula is used to find the coordinates of the midpoint by averaging the x and y of the two points. This is written as:Midpoint FormulaDISTANCEYou can find distance by plugging the rise and run into the Pythagorean theorem as though they are the sides of a triangle. You can also use the distance formula, which is derived from the Pythagorean theorem. Distance FormulaPARTITIONING LINE SEGMENTSTo partition a line at a ratio, you need to divide the line into parts by that ratio.Example: If there is a 1:3 ratio, then one part will be ? the size of the original line and another part will be ? the size of the original line.There is a formula to represent this.Partitioning Segments FormulaAt a ratio of A:B, the formula is…ANGLESThe set of all points between the sides of the angle are referred to as the INTERIOR of the angleThe EXTERIOR of an angle is the set of all points outside the angleAn angle can be named by its VERTEX (middle point) or by any combination of points with the vertex in the middle of the name.Example: Angle R, Angle PRT, Angle TRPAngles can also be numberedACUTE angles measure greater than zero and less than 90 degrees.RIGHT angles measure 90 degrees.OBTUSE angles measure greater than 90 but less than 180.STRAIGHT angles measure 180 degreesThe middle of an angle is called the VERTEXPROOFS NOTESAll proofs have a statement and a reason for each step. Some reasons can include, definitions, postulates, theorems, properties, “given”, and “simplify”.All items in a two column proof need to be numbered to make a list.Paragraph proofs are considered informal (Two column proofs are formal)It is acceptable to use abbreviations during a proofA proof uses DEDUCTIVE REASONING to create logical stepsCorollary-The immediate consequence of a result already proven. These are things you pick up in the process of writing a proof that are a result of something else you did.Postulate-This is a statement that is taken to be true and does not need to be proven.Theorem-A theorem is a proven statement.CONDITIONAL STATEMENTS NOTESA conditional statement is written as PQ , which is said, “If P, then Q”The CONVERSE is written Q P and is the reversed order of the statement.The INVERSE is written ~ P ~Q and is composed of the negatives of both the hypothesis and conclusion.Truth value of PTruth value of QTruth value of statementTFFTTTFTTFFTA statement is only false if the conclusion is false and the hypothesis is not.BICONDITIONAL STATEMENTS are true both forwards and backwardsThis means their converse is also true. ................
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