# Tutorial:Practice Block IV

## Area Relations of Similar Geometric Figures

In this activity you are going to use the following tools and algebraic input. Make sure you know how to use each tool before you begin.

 a = 2 Segment With Given Length Regular Polygon Move

In this activity you will recreate the following worksheet for your students. It allows them to discover the special relationship between the area of squares whose side lengths are a, a/2, and 2a.

### Area Relations

1. Measure the side length of the three squares below. Compare the side length of the blue square to the side lengths of the red and green squares. Which relation can you find?
2. Calculate the areas of the three squares. Compare the area of the blue square to the areas of the red and green squares. Which relation can you find?
3. Formulate a conjecture that compares the side length and area of the blue square to these of the red and green squares.
4. Try to proof your conjecture.
Hint: Assume that the side length of the blue square is a and calculate the areas of the corresponding squares.

### Construction Steps

1. Start your construction in GeoGebra by creating the number a = 2.

2. Construct the blue square starting with Segment With Given Length a. Then, use the two endpoints of this segment to create a regular polygon with 4 vertices.

3. In the same way construct the red square with side length a/2 and the green square with side length 2a.

4. Rename the vertices and change the properties of the squares (e.g. color, line thickness).

5. Prepare the GeoGebra window for the export of the graphics view as a picture (e.g. rearrange the squares, reduce the size of the GeoGebra window).

6. Export the graphics view as a picture and save your picture file.
Hint: Since your students are supposed to measure the side length of the squares you should not change the scale of the picture!

7. Open a word processing document and type in the heading and tasks of the worksheet.

8. Insert the picture of the squares into the worksheet.
Hint: Make a printout of the worksheet and try it out by measuring the side lengths of the squares.

### Challenge 1

Create similar examples for different geometric shapes (e.g. circle with given radius, equilateral triangle, rectangle). For which of these shapes do the same relations between the length of the given side (radius) and the resulting area apply? Try to find an explanation for this relationship between the given length and the area of the figure.

### Challenge 2

Create a dynamic worksheet based on your construction that helps your students to generalize their conjecture about the relationship between the side length and the area of such geometric shapes (for example, see Area_Circles.html).

## Visualizing the Angle Sum in a Triangle

In this activity you are going to use the following tools. Make sure you know how to use each tool before you begin.

 Polygon Angle Slider Midpoint Rotate Object around Point by Angle Move Insert Text

### Construction Steps

1. Create a triangle ABC.
Hint: Use counterclockwise orientation.

2. Create the angles α, β, and γ of triangle ABC.

3. Set the number of decimal places to 0 (Options Menu).

4. Create sliders δ and ε with the settings angle (type); 0° to 180° (interval); 10° (increment).

5. Create midpoint D of segment AC and midpoint E of segment AB.

6. Rotate the triangle around point D by angle δ (setting clockwise).

7. Rotate the triangle around point E by angle ε (setting counterclockwise).

8. Move both sliders to show 180° before you create the angles ζ (A’C’B’) and η (C'1B'1A'1).

9. Enhance your construction using the Properties Dialog.
Hint: Congruent angles should have the same color.

### Challenge 1

Insert dynamic text showing that the interior angles add up to 180°.
Hint: reate dynamic text for the interior angles (e.g. α = and select α from Objects), calculate the angle sum using sum = α + β + γ and insert the sum as a dynamic text.

Match colors of corresponding angles and text. Fix the text in the Graphics View.

### Challenge 2

Export the figure to a dynamic worksheet. Come up with instructions that guide your students towards discovering the angle sum in a triangle. Have them check their conjecture using the provided worksheet.

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