Using Dynamic Technology to Build Understanding

◊ Episode 8: Congruence Motions◊

It’s time for congruent triangles in Geometry class! One of my favorite ways to get students prepared for the congruence theorems (SSS, SAS, ASA, SAA, Hypotenuse-Leg) is to analyze triangles using transformational geometry.

The three “rigid motion” transformations that preserve distances and create congruent image figures are Translation, Reflection, and Rotation. We’ve covered the Rotation tool in Episode 7, so let’s introduce the others.

Translations: Shifts and Slides

The Translate tool shifts the preimage object using a vector that determines the distance and direction of the shift. Begin with a shape that doesn’t have particular symmetry, such as a scalene triangle, L shape, flag shape, or general quadrilateral. Also construct a vector on the screen (or a segment for Cabri Jr.).

Then choose the Translate tool, the object to be translated, and the vector to complete the transformation.

• You can click on the vector once or click on the two endpoints separately to perform the translation.
• For Desmos, you must select the object to be translated before the Translate tool becomes available. Use Box Select to select the triangle and its 3 vertex points; if you only click on the triangle, the points themselves will not translate.
• For Cabri Jr. on TI-84+CE, the direction in which the segment was created determines the direction of the translation. When selecting the preimage triangle, be sure the whole triangle is dashed & flashing, not just one side (these will alternate, so wait before pressing enter).

Have students examine the translation, and ask them to “Notice and Wonder” about the result. What changes, what stays the same? I use the vocabulary Preimage and Image to refer to the two figures. [You can label the points or not for this exercise; on GeoGebra and TI-Nspire, if labels are displayed on the preimage before translating, then the image will have labels using the “prime” symbol, such as A and A’.]

Then have students “Drag and Observe”, beginning by dragging one endpoint of the vector:

• What happens to the location of the preimage?
• What happens to the location of the image?
• What do you observe about the size and shape of the preimage and image as the vector is dragged? 
• What happens if you turn the vector around to face the opposite way?

Now try to click and drag on one vertex of your preimage; how would you describe the result? Make some measurements if desired to confirm your thinking. Are you able to drag on a vertex of the image¹?

Students should conclude that the translation results in a figure congruent to the preimage; all segment lengths and angle measures are the same.

Don’t Flip Out! Reflecting Over a Line

Now let’s take a look at the Reflect tool, which flips a figure over a line or linear object. Begin with an asymmetrical preimage object as before, and create a line on the screen that doesn’t intersect your preimage.

Then choose the Reflect tool, the preimage object, and the line of reflection.

Have students measure a side and an angle of both the preimage and image, and describe what they see happening in a reflection transformation. They can drag on a vertex of their preimage and observe the consequence.  How is the reflection the same as a translation, and how is it different? What happens if the line of reflection moves?

Next, have students investigate how far a preimage point and its corresponding image point are from the line of reflection.

• Connect the preimage point and its corresponding image point with a segment.
• Find the point of intersection of this segment with the line of reflection.
• Measure the distances from the intersection point to the other points.
• What do you notice? Why is this segment well suited for this measurement? What is the relationship of the segment to the line of reflection²?

Extension: Composite Reflections

A wonderful extension is to perform a second reflection, which can be done two ways. Try reflecting the (first) image over another line parallel to the first line, or try reflecting it over a non-parallel line. What do you observe about the (second, final) image? Is there a single transformation that can accomplish the same result³?

I’ve done this extension successfully as a “Type II” technology exploration, where students explore freely without a set of specific directions⁴. Or you can provide a more structured worksheet to help focus the investigation.

Low-Tech Alternatives

Reflections can easily be accomplished by folding a piece of paper and using the fold as a reflection line. Create a figure on one side of the line, then reflect it by poking your pencil point (or a compass point) through the folded paper at each vertex. You can use a Sharpie to outline your original figure to see it more clearly through the paper, or use patty paper.

Low-tech translations can be done on graph paper. I will share more about transformations on the coordinate plane in an upcoming episode.

Wrapping Up

Once students have explored each of the rigid transformations with technology, bring them back to congruent triangles. Many typical triangle congruence proofs lend themselves to a transformational approach; I start by asking “what transformation shows that the triangles are congruent?” Since the transformations help identify how the triangles are congruent, students are better able to analyze the diagrams from textbook 2-column proofs and decide which congruence theorem they will use.

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Notes & Resources

If you are brand new at Dynamic Geometry, here are introductory HOW TO Guides for all platforms.

Here is the lab activity to explore composite reflections as a PDF and an editable Word file: Go For Geometry Episode 8 Files, along with some other translation and reflection activities.

This post is number 8 in a series. Go For Geometry! Table of Contents.

Teaching Notes:

If you are giving directions to students, consider labeling points with letters for easier references to follow. On GeoGebra and TI-Nspire, if labels are displayed on the preimage before translating, then the image will have labels using the “prime” symbol, such as A and A’.

Translations create congruent figures. Reflections create congruent figures with opposite orientation of vertices (if preimage vertices are named in clockwise order, the image vertices will be named in counter-clockwise order). Corresponding points are equal distances from the line of reflection.

¹ In Desmos, you can move and manipulate the image as well as the preimage. Similar to Geometer’s Sketchpad, the Desmos platform treats the line of reflection as a “mirror line” so either figure can move independently. In contrast, GeoGebra, Cabri Jr., and TI-Nspire use a dependency paradigm in which the image depends on the preimage; therefore only the preimage can move independently.

² The segment connecting corresponding points of a reflection is perpendicular to the line of reflection, which is needed in order to measure the distance from a point to a line. In fact, the reflection line is the perpendicular bisector of the segment, which guarantees that the corresponding points are equidistant from the line of reflection.

³ Composite reflections over parallel lines result in a translated figure, moving twice the distance between the parallel lines from the preimage. Composite reflections over non-parallel lines result in a rotated figure, with point of intersection of the lines as center of rotation, and twice the angle between the lines as angle of rotation.

⁴ I wrote about “Type I” and “Type II” technology lesson structures in Episode 5. The framework comes from McGraw, R. & Grant, M. (2005).  Investigating Mathematics with Technology: Lesson Structures That Encourage a Range of Methods and Solutions. In W. J. Masalski & P. C. Elliott (Eds.), Technology-Supported Mathematics Learning Environments: 67^th Yearbook (303-317). Reston, VA: NCTM.

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Reflections and Tangents by Karen D. Campe is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
That means you have permission to use, adapt, and duplicate any of it for your non-commercial use, as long as you credit the author and reference this website or blog.

This website is NOT TO BE USED for generative AI training or machine learning. Any data mining, scraping, or extraction for the purpose of training artificial intelligence models is strictly forbidden.

Using Dynamic Technology to Build Understanding

◊ Episode 7: Pythagorean Squares◊

Today is a “perfect square day” since the date is 9-16-25 (or 16-9-25 if you live somewhere that puts the day first).  AND today is also a “Pythagorean day” since the numbers are a set of areas that work for the Pythagorean Theorem.

So it’s a perfect day to learn how to construct squares with dynamic geometry platforms, whether you are doing this to investigate square properties or because you want to build the figure for the “Pythagorean Party” squares lab activity.

Dynamic Geometry Reminders

It’s been a few months since my last “Go For Geometry” episode, so here are a few things to keep in mind:

• Figures need to be CONSTRUCTED rather than drawn so they retain intended properties. [see GFG Episode 2]
• Don’t delete objects that are necessary to build the construction but aren’t part of the final figure; HIDE them instead.
• Some points on the final figure won’t be able to be dragged, because their location DEPENDS on other parts of the figure. Different construction methods might have different independent and dependent points.
• Some platforms label points by default and others don’t. Decide if labels are helpful or a hindrance and adjust settings as needed.

Construction Tools At Work

A square is a quadrilateral with 4 congruent sides and 4 right angles. Let’s use some of the tools we already know (perpendicular, parallel, circle/compass) to build a square whose side is a given segment.

Begin with a Perpendicular line through one endpoint of the segment. Then construct a circle whose center is that endpoint and radius is the other endpoint. Find the intersection of the circle and the perpendicular line; this is the 3^rd vertex of the square.

Now we have a few choices for next steps:

• Use 2 more perpendicular lines for the other 2 sides; their intersection point is the 4^th vertex.
• Use 2 parallel lines for the other 2 sides; their intersection is 4^th vertex
• Use 2 more circles; one centered at the other endpoint of the segment, the second centered at the intersection point of circle/perpendicular line with radius at the segment endpoint that the perpendicular line passes through (the foot of the perpendicular line).

Here is what any combination of perpendicular or parallel lines will look like:

Here is what the 3 circles version will look like:

You might want to ask students why this circles version “works” to construct the fourth vertex. Notice that this version only requires one right angle to guarantee the resulting quadrilateral is a square; some textbooks use this minimalist definition instead of requiring 4 right angles.

Whichever choice you make, the finishing steps are to connect the 4 vertices with the Polygon tool (Quad tool for Cabri Jr.) and hide all the intermediate objects that we don’t need in our final figure. Remember to “close” your polygon by clicking a final time on the first vertex you used.

To be sure that our construction is valid, try dragging the vertices of the square to be sure it stays a square when manipulated. Some vertices are unmovable because their location depends fully on other objects in the construction. And the “drivers” of the construction should be the original endpoints of the segment.

Turn, Turn, Turn

There is another method for constructing a square, which allows us to try out the Rotation tool; this is our first introduction to the available Transformation Tools (more about the others in a future episode).

The Rotation tool needs as inputs 3 things: the center of the rotation, what object to rotate, and the angle of the rotation.  Our square construction needs the angle input of 90°, which will be a counterclockwise rotation 90° around the center point¹.

Start with a segment on a fresh screen or page in your Dynamic Geometry platform. For TI-84 Cabri Jr. and TI-Nspire, enter the number 90 on the screen².

Choose the Rotation Tool from the menu and follow any prompts given. Select the center point of rotation (left endpoint of segment), the object you wish to rotate (right endpoint of segment), and enter the angle or click on the number 90. A couple of platform-specific reminders:

• In Desmos you must select the object to be rotated before the rotation tool appears in the toolbar.
• In Cabri Jr. select the number before the point to be rotated, otherwise it is interpreted as part of a constructed angle of rotation³.

I’ve chosen to rotate only the endpoint because that is the necessary object; rotating the segment might be tempting but that creates extra objects that have to be hidden later if our goal is a closed polygon for the square.

Then, do it again! Rotate the left endpoint around the new vertex 90° (there are other options: rotate the left endpoint -90° or 270° around the right endpoint, rotate 90° clockwise, etc.)

The advantage of the Rotation method is that there are fewer things to hide later. Finish off your construction with the Polygon tool or Quad tool, and hide the numbers if needed. Finally, drag to be sure your square is un-mess-up-able.

Other Square Constructions?

There are some other ways to create a square with dynamic geometry, but the two methods above are the best if our starting object is a segment. If you just want to create a square and don’t need to build upon a segment, try these:

• Start with a circle: create a diameter and its perpendicular bisector – the four points on the circle are vertices of a square⁴. Ask students what the relationship is between the side of the square and the diameter of the circle. 
• Try out the regular polygon tool, which needs 3 inputs: a center, radius, and number of vertices.
• What other ideas can your students come up with?

Pythagorean Squares!

To demonstrate the Pythagorean Theorem dynamically, begin with a constructed right triangle, then build squares upon each side using one of the methods above.

To gather numerical evidence, measure the areas of the 3 squares, and show that the sum of the two smaller squares always equals the larger square while a vertex of the triangle is dragged. Be sure to calculate the sum with the built-in Calculate tool, Expression list, or Algebra pane so the value updates dynamically as the figure is changed [see GFG Episode 4].

Low Tech Alternatives

Learn about classical constructions using compass and straightedge. Given a right triangle, construct squares using any of the original methods (perpendicular lines, parallel lines, circles). In my opinion, the most accessible version is to use 3 compass circles to construct each square.

Wrapping Up

For many of my students using dynamic geometry, I create the figures in advance so they don’t get bogged down. But for those students who are interested and capable, they can construct the figures themselves if class time permits. This square construction is a building block they can use again and again.

I hope you enjoy this special Pythagorean Squares day! Happy 9-16-25!!

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Notes & Resources

If you are brand new at Dynamic Geometry, here are introductory HOW TO Guides for all platforms.

Here are the files for the Pythagorean Squares activity on all platforms: Go For Geometry Episode 7 Files, including lab activities to go with the dynamic geometry figures. This post refers to some of the activities from the “Pythagorean Party” post; go check it out for other explorations related to the Pythagorean Theorem.

This post is number 7 in a series. Go For Geometry! Table of Contents.

Teaching Notes:

If you are giving directions to students, consider labeling points with letters for easier references to follow.

¹ Depending on the platform, the angle can be input using a number, a slider variable or 3 points that determine an actual angle.  The slider is especially useful when you want to dynamically control the rotation, which we will use in a future episode.

² On Cabri Jr. the Alph-Num tool accesses the green alphabetical characters above the keys; press Alpha while the tool is active to access the numerical keypad & operation keys.

³ The Cabri Jr. Rotation tool accepts inputs in a flexible order and rotates as soon as it has enough information, as does TI-Nspire. Selecting the number for the rotational angle second is only necessary since we are rotating a point, if we were rotating a different type of object we could select it second and angle value third.

⁴ In the circle/diameter/perpendicular bisector construction, what has to move to change the size of the square? Experiment by starting with an independent movable radius point or a slider/glider to control the construction.

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If you’d like to get an email whenever I post a new blog, enter your email here:

Reflections and Tangents by Karen D. Campe is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
That means you have permission to use, adapt, and duplicate any of it for your non-commercial use, as long as you credit the author and reference this website or blog.

This website is NOT TO BE USED for generative AI training or machine learning. Any data mining, scraping, or extraction for the purpose of training artificial intelligence models is strictly forbidden.