Lesson 3 1 Core Focus on Geometry Reflections

Lesson 3. 1 Core Focus on Geometry Reflections

Warm-Up 1. What is the distance between the points (2, 7) and (− 1, 0)? Round to the nearest tenth, if necessary. ≈ 7. 6 units 2. Draw a figure with exactly one line of symmetry. Use a dashed line to show the line of symmetry.

Lesson 3. 1 Reflections Reflect an image on a coordinate plane.

Vocabulary Transformation Movement of a point or figure that changes its size or position. Pre-Image Original figure before a transformation. Image Resulting figure after a transformation. Reflection A type of transformation that flips a figure over a line. The pre-image and image are mirror images of each other.

Explore! Mirror, Mirror Step 1 Fold a piece of patty paper or tracing paper down the middle creating a vertical line. Open the paper and write your first name on one side of the line in uppercase letters. Step 2 Create a reflection of your name. Explain to a classmate how you created your reflection. Step 3 On a sheet of grid paper, graph a triangle at the top. Your vertices should have integer coordinates. This is your pre-image. Step 4 Draw a horizontal line on a grid line below your pre-image. This is your line of reflection. Create the reflection of your triangle below the line WITHOUT folding the grid paper. Explain to a classmate how you created your image. Check your solution by folding the grid paper on the line of reflection. Step 5 Are the pre-image and the image congruent? When working with polygons, do reflections always create congruent figures? Explain your reasoning.

Good to Know! The vertices of a pre-image are often labeled with letters. After a transformation occurs, the new image has vertices that are labeled with the same letter but an apostrophe is added. For example, if ABC is reflected over a line, then the image is labeled A B C. This is read “Triangle A prime, B prime, C prime. ” In mathematics, most transformations occur on a coordinate plane. The most common lines of reflection on a coordinate plane are the x- and y-axes. In the graph above, the line of reflection is the y-axis.

Example 1 The point D (− 3, 4) is reflected over the x-axis. What are the coordinates of its image? Graph the point on a coordinate plane. Reflect the point over the x-axis. Point D is 4 units above the x-axis. Point D is 4 units below the x-axis. The coordinates of the image are D (− 3, − 4).

Good to Know! Transformation rules describe the effect of a transformation on an ordered pair. y-axis reflection x-axis reflection When a figure is reflected over the y -axis, the x-coordinate changes sign. When a figure is reflected over the x -axis, the y-coordinate changes sign.

Example 2 R Graph the image of ∆RST under the transformation rule (x, y) → (x, −y). Which axis is the image reflected over? T Record the ordered pairs. R(4, − 3) S(4, − 1) T(1, − 1) Change each of the y-coordinates to its opposite. R (4, 3) S (4, 1) T (1, 1) Graph the ordered pairs of the image. The x-axis is the line of reflection. When the y-coordinates change to their opposites, the x-axis is the line of reflection. S

Reflections Over the x- and y- Axes When a point is reflected over the x-axis, the xcoordinate stays the same and the y-coordinate is changed to its opposite. (x, y) → (x, −y) When a point is reflected over the y-axis, the ycoordinate stays the same and the x-coordinate is changed to its opposite. (x, y) → (−x, y)

Example 3 A square has coordinates M(− 4, 2), N(− 1, 2), P(− 1, − 1) and Q(− 4, − 1). The square is reflected over the y-axis. What are the coordinates of the image? A reflection over the y-axis causes the y-coordinates to stay the same. M(− 4, 2) M ( 4 , 2 ) The x-coordinates change to their opposites. P(− 1, − 1) P ( 1 , − 1) N(− 1, 2) N ( 1 , 2 ) Q(− 4, − 1) Q ( 4 , − 1 ) The coordinates of the image are M’(4, 2), N’(1, 2), P’(1, − 1) and Q’(4, − 1).

Communication Prompt What are some examples of “reflections” that you encounter on a daily basis? How does a reflection in mathematics connect with the way we use reflections outside of a math classroom?

Exit Problems 1. The point (3, − 5) is reflected over the x-axis. What are the coordinates of the new point? (3, 5) 2. The triangle P(1, − 4), U(5, 0) and N(2, 1) is reflected over the y-axis. Graph the pre-image and image. Label all points.