7 1 Rigid Motion in a Plane Preimage

7. 1 – Rigid Motion in a Plane • Preimage, Image • P P’ • G: P P’ notation in relation with functions. • If distance is preserved, it’s an isometry. • An isometry also preserves angle measures, parallel lines, and distances. These are called rigid transformations. • Examples: Shifting a desk preserves isometry. Projection onto a screen normally doesn’t • Superimposition – If you can put one figure on top of another by a rigid transformation, then the two figures are CONGRUENT.

Reflections When a transformation occurs where a line acts like a mirror, it’s a reflection. P Q m P` R=R` Q`

Translations When a transformation occurs where all the points ‘glide’ the same distance, it is called a TRANSLATION. P (-5, 3) (-3, 2) (-4, 1)

Rotations. A rotation is a transformation where an image is rotated about a certain point. P O Positive, counterclockwise Negative, clockwise

We’ll describe the transformations and line up some letters.

You can show that something is an isometry on a coordinate plane by using distance formula. Show using distance formula which transformations are isometric and which aren’t.

• You will have some problems where you need to match up parts, make them equal to each other, and solve. What you want to do is match up corresponding parts.

7. 2 – Reflections When a transformation occurs where a line acts like a mirror, it’s a reflection. A reflection in line m maps every point P to point P’ such that 1) If P is not on m, then m is the perpendicular bisector of PP` 2) If P is on line m, then P`=P P m P` Q R=R` Q` Line of Reflection Notation Rm: P P` Name of line the transformation is reflecting with.

Write a transformation that describes the reflection of points across the x-axis (0, 5) (-3, -1) (1, -3) Sometimes, they want you to reflect across other lines, so you just need to count.

Theorem 14 -2: A reflection in a line is an isometry. Therefore, it preserves distance, angle measure, and areas of a polygon. Sketch a reflection over the given line.

Hit the black ball by hitting it off the bottom wall. Use reflection! Reflection is an isometry, so angles will be congruent by the corollary, so if you aim for the imaginary ball that is reflected by the wall, the angle will bounce it back towards the target.

This concept also occurs in the shortest distance concept Where should the trashcan be placed so it’s the shortest distance from the two homes. Shortest distance is normally a straight line, so you want to mark where the shortest path would be from the two different homes by using reflection. Anywhere else will give you a longer path.

Given A (4, 2) and B (-5, 4), find a point C on the x-axis so that AC + BC is a minimum. ? ? ? Like the housepool table problem, trying to find the shortest distance.

A figure in the plane has a line of symmetry if the figure can be mapped onto itself by a reflection in the line. We think of it as being able to cut things in half.

Sketch and draw all the lines of symmetry for this shape

7. 3 –Rotations. A rotation is a transformation where an image is rotated about a certain point. RO, 90: P P` P O Amount of rotation. Point of P` rotation Fancy R, Positive, counterclockwise rotation Negative, clockwise

As you may know, a circle is 360 degrees, so if an object is rotated 360, then it ends up in the same spot. RO, 360: P P` P` Then P = P` P O Likewise, adding or subtracting by multiples of 360 to the rotation leaves it at the same spot. RO, 60: P P` =RO, -300: P P` =RO, 780: P P` O

A rotation about point O through xo is a transformation such that: 1) If a point P is different from O, then OP`=OP and 2) If point P is the same as point O, then P = P` Thrm: A rotation is an isometry. P O xo P`

Find image given preimage and rotation, order matters RJ, 180: ABJ RN, 90: IJN RN, -90: IJN RNF: IMH RD, -90: FND

Figure out the O = Origin coordinate. RO, 180: P P` O = Origin RO, 90: (2 , 0) ( ) RO, -90: (0 , 3) ( ) RO, 90: (1 , -2) ( ) RO, -90: (-2 , 3) ( ) RO, 90: (x , y) ( ) RO, -90: (x , y) ( )

Rm: P P` P O n Rn: P` P`` The composite of the two reflections over intersecting lines is similar to what other transformation? m Theorem Referencing the diagram A composite of ______ in two above, how much does ______ lines is a _____ P move by? about the point of intersection of the two lines. The measure of the angle of ____ is twice the measure of the angle from the first line of reflection to the second. Justify?

Find angle of rotation that maps preimage to image 18 o 70 o

7. 4 – Translations and Vectors When a transformation occurs where all the points ‘glide’ the same distance, it is called a TRANSLATION. Notation T: P P` T for translation Generally, you will see this in a coordinate plane, and noted as such: T: (x, y) (x + h, y + k) where h and k tell how much the figure shifted. Theorem: A translation is an isometry.

We will take a couple points and perform: T: (x, y) (x + 2, y – 3) T: (2, 3) ( __ , __ ) T: ( , ) (5, -1) T: (-3, 0) ( __ , __ ) T: ( , ) (0, 1) T: (a, b) ( __ , __ ) T: ( , ) (c, d)

Vectors Any quantity such as force, velocity, or acceleration, that has both magnitude and direction, is a vector. AB Vector notation. ORDER MATTERS! B Initial A AB Terminal Component Form

Write in component form AB CD C B D A

Translations You could also say points were translated by vector (-5, 3) (-3, 2) (-4, 1) Translate the triangle using vector

Write the vector AND coordinate notation that describes the translation ‘ ‘

Rm: P P` P m Theorem Rn: P` P`` P` The composite of the two reflections over parallel lines is similar to what other transformation? n Referencing the diagram above, how far apart are A composite of _____ in P and P``? two ____ lines is a _____. The _____ glides all points through twice the distance from the first line of reflection to the second.

M and N are perpendicular bisectors of the preimage and the image. How far did the objects translate ABC translated to ______ m n ------4. 2 in ----

7. 5 – Glide Reflections and Compositions A GLIDE REFLECTION occurs when you translate an object, and then reflect it. It’s a composition (like combination) of transformations.

We will take a couple points and perform: T: (x, y) (x + 2, y – 3) Ry-axis: P P` Then we will write a mapping function G that combines those two functions above. T: (2, 3) ( __ , __ ) Ry-axis: ( __ , __ ) T: (-3, 0) ( __ , __ ) Ry-axis: ( __ , __ )

Order matters in a composition of functions. The composite of two isometries is an isometry. There are times on a coordinate grid where you’ll be asked to combine a composition into one function, like you did for glide reflections. There also times when two compositions may look like a type of one transformation. Remember, if you can describe a set of rigid transformations that will place one object on top of another, then the two objects are congruent by superimposition.

We’ll do different combinations of transformations and see what happens.

We’ll do different combinations of transformations and see what happens. Points and Shapes. Also draw some transformations and describe compositions, this will illustrate the concept of superimposition.

Find image given preimage, order matters

8. 7 – Dilations. A dilation DO, k maps any point P to a point P`, determined as follows: 1) If k > 0, P` lies on OP and OP` = |k|OP 2) If k<0, P` lies on the ray opposite OP and OP` = |K|OP 3) The center is its own image Center Scale factor P O |k| > 1 is an EXPANSION, expands the picture |k| < 1 is a CONTRACTION, shrinks the picture

B O A C

When writing a scale factor of a dilation from O of P to P’, the scale factor is: Does dilations or similarity mapping allow you to prove congruence by superimposition? Is it a rigid transformation?

Identify scale factor, state if it’s a reduction or enlargement (double checking), find unknown variables. O 2 30 o A B 6 4 2 x C B’ 4 y A’ (2 z)o Find x, y, z 18 C’

Identify scale factor, state if it’s a reduction or enlargement (double checking), find unknown variables. AA’ = 2 A’O = 3 20 2 y A A’ 12 x B’ 60 o (3 z)o O D’ D B C’ C

DO, 2 DO, 1/3 1) Dilate each point by scale factor and label. 2) Connect

• Find other sides, scale factor, given sides of one triangle, one side of another