Math 416 Geometry Isometries

Topics Covered 1) Congruent Orientation – Parallel Path 2) Isometry 3) Congruent Relation 4) Geometric Characteristic of Isometry 5) Composite 6)Geometry Properties 7) Pythagoras –

Congruent Figures Any two figures that are equal in every aspect are said to be congruent Equal is every aspect means… –All corresponding angles –All corresponding side lengths –Areas –Perimeters

Congruent Figures We also note that we are talking about any figures in the plane not just triangles However, it seems in most geometry settings, we deal with triangles We hope this section will allow you to look at all shapes but…

Orientation One of the most important characteristics of shapes in the plane is its orientation How the shape is oriented means the order that corresponding point appear

Orientation Consider #1#2 A A’ B C B’ C’

Orientation To establish the order of the points, we need two things; #1) A starting point – that is a corresponding point #2) A direction – to establish order “Consistency is the core of mathematics” I will choose A and A’ as my starting points

Orientation I will choose counterclockwise as my direction Hence in triangle #1 we have A -> B > C. In Triangle #2 we have A’ -> B’ -> C’

Orientation Since the corresponding points match, we say the two figures have the same orientation. Consider A A’ #1 #2 B C C’ B’

Orientation Vocabulary These figures do not have the same orientation Same orientation can be phrased as follows; – orientation is preserved - orientation is unchanged - orientation is constant

Orientation Vocabulary Different Orientation can be stated - orientation is not preserved - orientation is changed - orientation is not constant

Parallel Path We are interested how one congruent figure gets to the other We are interested how one congruent figure is transformed into another We call the line joining corresponding points its path i.e. A A’ is the path If we look at all the paths between corresponding points, we can determine if all the paths are parallel.

Examples These are a parallel path A C’ C B’ B A’

Examples A B C B’ A’ C’ These are not parallel paths It is called Intersecting Paths

Types of Isometries There are 4 Isometries 1) Translation 2) Rotation 3) Reflection 4) Glide Reflection

Translation Translation – moving points of a figure represented by the letter t. As you may recall t (-2,4) (x – 2, y + 4) You move on the x axis minus 2 and on the y axis you move plus 4.

Rotations Rotations: Rotations can be either 90, 180, 270 or 360 degrees. Rotations can be clockwise or counter-clockwise Represented by the letter r

Reflection You can have reflections of x You can have a reflections of y

Glide Reflection Glide reflection occurs when the orientation is not preserved AND does not have a parallel path. Can be best seen with examples…

Tree Diagram We can define the four isometries by the way of these two characteristics Orientation Same? Parallel Path? YES No TRANSLATION ROTATION REFLECTION GLIDE REFLECTION

Table Representation Orientation Same (maintained) Orientation Different (changed) With Parallel Path TranslationReflection Without Parallel Path RotationGlide Reflection

Notes The biggest problem is establishing corresponding points. It is easy when they tell you A  A’, B  B’ but it is usually not the case Let’s try two examples… what kind of isometric figures are these… You may choose to cut up the figure on a piece of paper which can help locate the points…

Example #1 Consider (we assume they are congruent) We need to establish the points. Look for clues (bigger, 90 and smaller angle). 90° Bigger Angle Smaller Angle

Which Isometric Figure? Hence orientation ABC A’C’B’ are NOT the same… Parallel paths… No! A C’ C B’ B A’ GLIDE REFLECTION ORIENTATION? PARALLEL PATH?

Example #2 A B’ C B C’ A’ ABC and A’B’C’ – Orientation the same ORIENTATION? PARALLEL PATHS? Not Parallel Paths ROTATION

Other Figures When the figure is NOT a triangle, you can usually get away with just checking three points. The hard part is finding them. Let’s take a look at two more examples

Example with a Square ° ° B C B’ C’ A’ A Orientation / Parallel Paths? Orientation Changed, Not Parallel Glide Reflection

Practice ° ° Orientation? Parallel? Orientation Same; Not Parallel  Rotation 90 o counter clockwise rotation

The Congruency Relation When we know two shapes are congruent (equal), we use the symbol. Congruent Symbol

Congruency Relation Hence if we say HGIJ KLMN We note H corresponds to K G corresponds to L I corresponds to M J corresponds to N

Congruency Relation From this we state the following equalities. Line length HG = KL (1 st two) GI = LM (second two) IJ = MN (last two) HJ = KN (outside two)

Congruency Relation Angles < HGI = < KLM (1 st two) < GIJ = < LMN (second two) < IJH = < MNK (last two first) < JHG = < NKL (last one 1 st two) We have established all this without seeing the figure!

Exam Question State the single isometry. State the congruency relation and the resulting equalities. A D C B K L N M Hence BACD KMNL

Exam Question We can also can note that… B  K D  L C  N A  M Clockwise Orientation / Parallel Path?

Exam Solution Orientation Changed Parallel Path Reflection

Other Findings Line Length BA = KM AC = MN CD = NL DB = LK Angles < BAC = < KMN < ACD = < MNL < CDB = < NLK < DBA = < LKM

Test Question Given ABCDE FGHIJ True or False? You should draw a diagram to clarify… False A DC B I G J H F < ABC = HIJ E < ABC = HGF True BC = HI False

Two Isometries – Double the fun! At certain points, we may impose more than one isometry. Consider 12 We say 1  2 is a reflection of s 3 Math #1 2  3 is a rotation r

Notes We would say that the composite is r ° s after We can say there is a rotation after a reflection. So you should read from right to left

Notes We also note that 1 – 3 is a glide reflection (gr) Hence r ° s = gr

Practice Consider  2 t 2  r Thus r ° t = r Math is fun

Geometry Reminders Complimentary Angles Here are some reminders of things you should know. b a Complimentary angles add up to 90 o. Thus <a + <b = 90 o

Supplementary Angles a b Supplementary angles add up to 180 o. All straight lines form an angles of 180 o. Thus <a + < b = 180 o

Vertically Opposite Angles a b d c Vertically opposite angles are equal. Thus <a = <c and <b = <d

Isoscelles Triangles The angles opposite the equal sides are equal or vice versa xx

Angles in a Triangle a bc Angles in a triangle add up to 180 o. Thus <a + <b + <c = 180 o.

Parallel Lines a b c d x w yz When a line (transversal) crosses two parallel lines, four angles are created at each line Transversal Line

Parallel Lines The following relationship between each group is created. Alternate Angles - both inside (between lines) & the opposite side of tranversal are EQUAL. Thus, < c = < x < d = < w a b c d x w yz

Corresponding Angles Both same side of tranversal one between parallel lines the other outside parallel lines are EQUAL <a = <w <c <y <b = < x <d = <z a b c d x w yz e <b & <e are called alternate interior angle

Supplemental Angles Both same side of transversal Both between parallel lines Add up to 180° Therefore, <c + <w = 180° <d + <x = 180°

Practice 5x+35 2x + 92 A D F C G H B We note < DEB = < ABG (corresponding) <DEB = <HEF (vertical) E 5x+35=2x+92 3x = 57 X = 19

130 A D F C G H B Solution Replace x = 19 into 5x+35 5(19) + 35 = 130

Test Question What is the angle < ABC? 5x x x + 5 = 180 8x -12 = 180 8x = 192 x = 24 Replace x = 24 into 2x – 20 2 (24) – 20 = 28° 5x+3 2x-20 x+5 A C B

Pythagoras Theorem The most famous and most used theorem or geometric / algebraic relationship is Pythagoras Theorum In words – the square of the hypotenuse is equal to the sum of the square on the of the other two sides

Pythagoras Example Which of these numbers (3,4,5) must be the hypotenuse? Establish 90° Does the placement of the 3, 4 or 5 make a difference? Formula c 2 = a 2 + b 2 Have one unknown. Solve and switch for practice

Pythagoras in Geometry If we have a right angle triangle with a 30° (or a 60°) The side opposite the 30° angle is half the hypotenuse Or.. the hypotenuse is twice the side opposite the 30° angle

Practice ½x x 30° Hence if the hypotenuse is 8, x = ? x = 4 or 2x x 30°

Practice 5 x 60° x = ? x = 10 y y = ? 10 2 = y = 25 + y 2 75 = y 2 y =8.66