Reflections Geometry

REFLECTION A reflection is a type of transformation that uses a line that acts like a mirror. The mirror line is the line of reflection. Each point of the original figure has an image that is the same distance from the line of reflection as the original point but is on the opposite side of the line.

REFLECTION The line of reflection is the perpendicular bisector of the segment joining every point and its image.

Definition of Reflection A reflection in a line m is a isometric transformation that maps every point P in the plane to a point P’, so that the following properties are true;   1. If point P is NOT on line m, then 2. If point P is ON line m, then

Vocab Perpendicular Bisector Perpendicular Bisector

Graph the Reflections Graph the reflection in a coordinate plane W(-3,3) in the y-axis Z(1,3) in the line x=1 What do you notice about the original point and the new point in question 1?

Reflections When reflecting in the y-axis, the y coordinate stays the same while the x coordinate changes signs. (x, y)  (-x, y)

Reflections When reflecting in the x-axis, the x coordinate stays the same while the y coordinate changes signs. (x, y)  (x, -y)

Reflections When reflecting in the line y = x, the y coordinate and x coordinate keep the same sign, but the order is reversed. (x, y)  (y, x)

Example Find the coordinates of the reflection without using a coordinate plane. S (0,2) reflected in the x-axis Q (-3,-3) reflected in the y-axis

Example Find the coordinates of the reflection without using a coordinate plane. T (3,8) reflected in the x-axis R (7,-2) reflected in the y-axis

Reflection When a reflection occurs, does the figure change shape or size? This means that a reflection is an __________.

Isometric Properties Distance (lengths of segments are the same) Angle Measure (angles stay the same) Parallelism (things that were parallel are still parallel) Collinearity (points on a line, remain on the line)

Example The coordinates of triangle ABC are A(1, 3), B(4, 2), and C(3, 6). If the triangle is reflected on the x-axis, what is the length of A’B’?

Transformation Properties Distances are different – Points in the plane move different distances, depending on their distance from the line of reflection. Orientation is reversed – In ▲ABC the points in a clockwise direction come in the order of A-B-C but the image comes in the order of A’-C’-B’ Special Points – Points on the line of reflection do not move at all under the reflection.