Parallelogram to Rectangle
Visual Proof in GeoGebra
Presented by Monika Bell, Suzy Giorgione, Jeff Rose
The purpose of this worksheet is to provide a step-by-step guideforcreating a construction that offers dynamic visual proof that any parallelogram can be turned into a rectangle with the same base and height. Students with GeoGebra experience may be able to move at a faster pace than those who are newer to the program. Please, don’t let us stop you. But, should you move ahead, please do not ask a question about step 15 when the class is on step 5. Also, if you finish your construction before the rest of the class, spend some time “tricking out” your construction, or helping other students who are not done (if they want your help!). These instructions are based on version RC 3 of GeoGebra 3.0 (11/20/07), which can be obtained by visiting and clicking on the WebStart link. Other versions may have tools in other icon locations – but that should be the only difference. Please click on the View tab and check the Algebra window option.
Here we go …
Stage 1 – Creating the Parallelogram
1)Using the tool Segment between two points(third icon from left), create a diagonal segment, . Create a second (horizontal) segment, . These will be adjacent sides of your parallelogram.
2)Click the Parallel line tool (fourth icon from left). Click Point B and then click .
3)Now click Point C and then click .
4)Select Intersect two objects (second icon) to create Point D. This is the intersection of the two parallel lines.
5)Select the Polygon tool (fifth icon) to fill in the parallelogram ACBD.
6)Select the Line through two points tool (third icon) and click on Points A and C to create .
7)Select the Perpendicular line tool (fourth icon). Make a perpendicular line by clicking Point B and then Line e.
8)SelectIntersect two objects to create Point E, which is the intersection between Line e and Line f.
9)Choose Segment between two points(third icon) to create . Right click this new segment and rename it “h”.
10) While in Move mode (first icon), double-click on any of the lines (in a spot that’s away from the parallelogram.) This will allow you to open the Properties menu. Click the word Line from the menu on the left, and select the Basic tab. Now, uncheck Show object and uncheck Show label. This hides all the lines.
11)Now click the word Segment. Under the Basic tab uncheck Show label. This hides all segment labels. We need to keep the labels for b and h, so go back and select those two segments and check Show label – Name and Value.
12)Click the word Quadrilateral from the menu. You can click the Color tab to personalize your parallelogram. We’re finished with the properties, so click on Done.
13)Make sure you are set to Move mode. Now click and drag any of the blue vertices to make sure your parallelogram is moveable.
Stage 2 – Translating the Triangle
14)Select the Slider tool (ninth icon). Name the slider “t” and set the min at 0 and the max at 1.
15)In the Input field (at the bottom), define Point F using the formula “F=A+t*(C-A)”. This creates a moveable point, F, from A to C controlled by Slider t. At this point, click on the View tab and uncheck the Algebra window to discontinue the automatic labeling feature.
16)Select a Vector between two points (third icon). Click Point A,thenPoint F.
17)Click the Polygon tool. Create triangle ABE.
18)Click the Translate object by vectortool (eighth icon). Click triangle ABEand then click the vector.
19)Right-click the vector and uncheck Show object to hide the vector. The triangle should now translate as you move Slider t.
Stage 3 – Conditional Showing and Hiding
20) Set Slider t to 1.
21)Right-click the original triangle to display its’ Properties. Click the Color tab and choose white. Click the Styletab and choose Filling100%. Click the Advanced tab, and type “t ≠ 0”.
22)Right-click the translated triangle to display its’ Properties. Click the Color tab a different color. Click the Advanced tab, and type “t ≠ 0”.
23)In the Input field, type “area = b*h” and hit Enter.
24)Click the Insert text tool (ninth icon) and click anywhere on your drawing pad away from your construction. Type “Area = Base* Height” and hit OK.
25)Click underneath that text and type the following formula, exactly: area + ” = “ + b + ” * “ + h and then hit OK to create text that is linked to the variables area, b and h.
26)Create a Check box (ninth icon) and in the Caption box type in “Area”. For Select object, you can either use the drop down menu to select Text text2, or click on the text itself.
27)Finally, in Move mode, double-click the Check box. Click the Advanced tab, and type “t =1”. This will make it so that the Check box will only appear when the student has moved the slider all the way over.
That’s it! At this point, you can play around with your construction to make it look the way you want it to. Suggestions include hiding all the points except A, B and C, or creating segments named l and w that correspond to the sides of the rectangle. After you’ve got this one down, it’s not hard to create more dynamic visual proofs of other geometric formulas and theorems. Have fun!