# TRANSFORMATIONS – REFLECTION EXPLORATION

- Author:
- Suzanne Harper

**Reflections in geometry have some of the same properties of a reflection you observe when looking into a mirror. In this activity, you’ll investigate the properties of reflections that make a reflection the “mirror image” of the original.**

## Part 1: Reflecting Triangles

*Predict*where the image of Δ

*CDE*will be once reflected over line AB. Construct points for vertices C’, D’ and E’ in your sketch. [

__Technology Tip__: Along the top of the sketch window, you will see a row of buttons, referred to as the

*Tool Bar*. Under the second button from the left, select

**Point**; and click anywhere in your sketch window to create a point.]

*CDE*(both its sides and vertices) over line

*AB*. [

__Technology Tip__: Along the tool bar, under the third button from the right, select

**Reflect Object about Line**; select the triangle CDE and the line of reflection.]

3. Assess your prediction points. How close were *your prediction points* to the (actual) image vertices of Δ*CDE*?

4. *Predict *the movement of the triangles when you __drag the vertices of the original ____ΔCDE__. Observe and describe how the triangles are related. Also, be sure to drag the line of reflection.

5. Are a figure and its mirror image always congruent under manipulation of the triangle or of the mirror? Why?

__Technology Tip__: Under the third button from the left along the top tool bar, select

**Segment**; then select corresponding vertices on the triangles. Remember to go back to the

**Selection Arrow**(the first button on the left) to end drawing segments. Click on the segment in your sketch, and look for a button on the top right corner of your sketch window with three segments and a purple triangle and circle. Click on that button, use the options on the

**Color**and

**Style**sub-buttons.]

7. Predict the relationships between the dashed segments and the mirror line. Drag the vertices and sides of the triangle around and observe the relationship.

8. [Challenge] Suppose GeoGebra didn’t have a **Reflection Tool**. How could you construct a given point’s
reflected image over a given line? Try it. Start with a point and a line (as shown in the sketch below). Come up with a construction for the reflection of the point over the line. Describe your method using both words and sketches.

## Part 2: Reflections in the Coordinate Plane

*x*- and

*y*- axes in the coordinate plane.

*ABC*with vertices on the grid. [

__Technology Tip__: You can use either the point tool and connect the points by segments; or you can use the segment tool to construct both your vertices and sides of your triangle.]

2. Record the coordinates of each of the vertices below.

3. Make a prediction where the image of Δ*ABC* would lie once reflected over the *y*-axis. Then reflect Δ*ABC* over the *y*-axis to assess your prediction. Record your results below.
*
Reflected over the y-axis prediction*: A′ = ______ B′ = ______ C′ = ______

*Reflected over the y-axis*: A′ = ______ B′ =______ C′ =

**actual****_______**

*y*-axis.

4. Describe any relationship you observe between the coordinates of the vertices of your original triangle and the coordinates of the reflected image across the *y*-axis.

5. Draw a new triangle in the sketch window below. Record the coordinates of each of the vertices below.

6. Make a prediction where the image of Δ*ABC* would lie once reflected over the *x*-axis. Then reflect Δ*ABC* over the *x*-axis to assess your prediction. Record your results below.
*
Reflected over the x-axis prediction*: A′ = ______ B′ = ______ C′ = ______

*Reflected over the x-axis*: A′ = ______ B′ =______ C′ =

**actual****_______**

*x*-axis.

7. Describe any relationship you observe between the coordinates of the vertices of your original triangle and the coordinates of the reflected image across the *x*-axis. Delete your triangle’s image.

8. Graph the line *y = x*, by typing in the equation in the **Input line**. Reflect your triangle across this line. Describe any relationship you observe between the coordinates of the original points and coordinates of their reflected images across the line *y = x*.
*Reflected over the line y = x ***actual**: A′= ______ B′ = ______ C′ = _______

9. Make some overall general statements about reflections. What were some of the main results from today's tasks.