14 – 2 Reflections The guy isn’t drowning, it’s just your reflection. P Q P` Q` Line of Reflection When you ponder the past. When a transformation occurs where a line acts like a mirror, it’s a reflection. R=R` 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 Notation R m : P  P` m Name of line the transformation is reflecting with. R for reflection

Theorem 14-2: A reflection in a line is an isometry. Therefore, it preserves distance, angle measure, and areas of a polygon. Therefore, those three things are INVARIANT under reflection. Orientation is NOT invariant, changes orientation, or direction in a sense.

Draw the reflection across the dotted line.

Write a transformation that describes the reflection of points across the x-axis R x-axis :(x,y)  (x, -y) (1, -3)(1, 3)(-3, 1) (0, 5)(0, -5)

Key to reflections is perpendicular bisectors. You will need to construct in your homework, this is how. Use construction to reflect PQ across line m m Construct line perpendicular to m from point P P Q Use compass, intersection as center, swing compass to other side. Make dot. Repeat, then connect dot. P` Q`

Hit the black ball by hitting it off the bottom wall. Use reflection! AIM HERE! 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.

14 – 3 Translations and Glide Reflections When a transformation occurs where all the points ‘glide’ the same distance, it is called a TRANSLATION. Bueno means good. 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. T: (x,y)  (x + 4, y + 2) (-5, 3)(-1, 5)(-3, 2)(1, 4)(-4, 1)(0, 3) You could also say points were translated by vector (4,2) P P`

Theorem 14-3: A translation is an isometry. Therefore, it preserves distance, angle measure, and areas of a polygon. Therefore, those three things are INVARIANT under reflection. Orientation is ALSO invariant for translations, because things are still oriented the same way.

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) R y-axis :P  P` Then we will write a mapping function G that combines those two functions above. T:(2,3)  ( __, __ ) R y-axis :( __, __ )  ( __, __ ) T:(-3,0)  ( __, __ ) R y-axis :( __, __ )  ( __, __ ) G:( x, y)  ( ____, ____ )

W - HW #68: Pg 580: 4—12 even, 13— 15, 21—22, 24, 26—30, 38 (17 problems) Pg 586: 2—7, 9, 11, 12 (9 problems)