4.2 Vocabulary Remember…Transformation, Preimage, Image, Isometry, Congruence Transformation… Reflection

A Reflection is a rigid transformation (isometry) that uses a line like a mirror to reflect a figure. The mirror line is called the line of reflection. The reflected image is congruent to the preimage, so reflections are a Congruence Transformation.

REFLECTIONS Draw a segment from each vertex of the preimage to the corresponding vertex of the image. Your construction should show that the line of reflection is the perpendicular bisector of every segment connecting a point and its image.

Check It Out! Example 2 Copy the quadrilateral and the line of reflection. Draw the reflection of the quadrilateral across the line.

In general, you can reflect over any line but the typical lines of reflection are: x = 0 (y-axis), y = 0 (x-axis) and y = x

Example 3A: Drawing Reflections in the Coordinate Plane Reflect the figure with the given vertices across the given line. X(2, –1), Y(–4, –3), Z(3, 2); x-axis The reflection of (x, y) is (x,–y). Y’ X(2,–1) X’(2, 1) Z X’ Y(–4,–3) Y’(–4, 3) X Z(3, 2) Z’(3, –2) Z’ Y Graph the image and preimage.

Example 3B: Drawing Reflections in the Coordinate Plane Reflect the figure with the given vertices across the given line. R(–2, 2), S(5, 0), T(3, –1); y = x S’ R’ T’ The reflection of (x, y) is (y, x). R(–2, 2) R’(2, –2) S R T S(5, 0) S’(0, 5) T(3, –1) T’(–1, 3) Graph the image and preimage.

Example 4: Problem-Solving Application Two buildings located at A(8,4) and B(4,0) are to be connected to the same point on the water line y = 10. Where should they connect C(x, y) so that the least amount of pipe will be used? B(4,0) . . A(8,4) y=10 C ‘

Understand the Problem Example 4 Continued 1 Understand the Problem The problem asks you to locate point C on the water line so that AC + CB has the least value possible. 2 Make a Plan Let A’ be the reflection of point A across the water line. For any point C on the water line, CA’ = CA so BC + CA = BC + CA’. BC + CA’ is least when B, C, and A’ are collinear.

C( 13 2 , 10) Example 4 Continued Solve Reflect A across the water line and find coordinates of A’. Using B and A’ write the equation for BA’. Then solve the system of equations to find the intersection of BA’ and the waterline. The connection point is? C( 13 2 , 10)

Reflect the figure with the given vertices across the given line. Lesson Quiz: 1. Tell whether the transformation appears to be a reflection. Reflect the figure with the given vertices across the given line. 2. A(2, 3), B(–1, 5), C(4,–1); y = x 3. U(–8, 2), V(–3, –1), W(3, 3); y-axis 4. E(–3, –2), F(6, –4), G(–2, 1); x-axis