Problem set
1. Construct an equilateral triangle. Construct the midpoints of each side. Join the midpoints. What can you say about the triangles that are formed? Are they isosceles, equilaterals, right, similar, congruent?
2. Make a script for an equilateral triangle. Use it to construct an outward equilateral triangle on each side of a square.
3. In the diagram produced in Problem 3, join counterclockwise the vertices of the triangles (with a segment) that are not vertices of the square. What kind of quadrilateral is formed?
4. Compare the constructions of a square and a rectangle (Examples 6 and 8). Use them to explain to your students that a square is also a rectangle.
5. In the construction of a rectangle (Example 8), Figure 35 shows only the measurement of three angles. Explain why this information implies that the fourth angle is also a right angle.
6. Construct 
and find the midpoints
D, E and F, of each side. Construct 
. Find the
areas of 
and 
. What can you say
about them? Are they related?
7. Constructing a parallelogram. A parallelogram is a quadrilateral with two pairs of parallel sides. Use this definition to construct a parallelogram.
8. Constructing a rhombus. A rhombus is a quadrilateral with four congruent sides. Use this definition to construct a rhombus. Argue that a rhombus is a parallelogram (Hint: Show that the alternate interior angles formed by, say, sides CD and BC, and the diagonal BD are congruent. Then conclude that sides CD and BC are parallel.)
9. Constructing midpoints and a midquad. "The midpoint quadrilateral (midquad) is the quadrilateral obtained by connecting consecutive midpoints of the sides of a given quadrilateral. Construct a quadrilateral ABCD and midquad EFGH." Drag each vertex of quadrilateral ABCD at a time. What happens to quadrilateral EFGH? What kind of a quadrilateral is it? Try to encompass in your answer all possible shapes of quadrilateral EFGH. Hint: Measure the angles!
10. In quadrilateral ABCD with midquad EFGH, measure the areas of each quadrilateral. Drag any of the vertices of quadrilateral ABCD. Are the areas related? And if so, how? (Hint: Taking the ratio of the areas would be very helpful.) Write your answer as a conjecture, i.e. complete the phrase: The area of the midquad seems to be...)
11. Construct a quadrilateral ABCD, and its midpoint quadrilateral. Construct the midpoint quadrilateral of the previous quadrilateral. Repeat this process two more times. Find the areas of each quadrilateral starting with quadrilateral ABCD. Use the built-in calculator to find the ratio of the areas of two consecutive quadrilaterals. Divide the area of the outside quadrilateral by the area of the inside quadrilateral. Do you observe a pattern? What type of sequence (arithmetic or geometric) do the ratios form?
12. Compare the constructions of a square and a rhombus (Example 6 and problem 7). Use them to explain to your students that a square is also a rhombus.
13. Compare the constructions of a square (Example 6), a rectangle (Example 8), a parallelogram (Problem 6), and a rhombus (Problem 7). Use them to explain that a square, a rectangle and a rhombus are parallelograms.
14. Constructing a trapezoid. A trapezoid is a quadrilateral with exactly one pair of opposite sides parallel. We use this property to construct a trapezoid. Use this definition to construct a trapezoid.
15. A trapezoid ABCD with parallel sides AB and CD is called isosceles trapezoid if segments BC and DA are congruent. The construction of an isosceles trapezoid is beyond the scope of this webmodule however, with the dragging power of the software you can get "close" to one. Here it is how. Construct a trapezoid ABCD with parallel sides AB and CD. Measure sides BC and CD. Drag both vertices C and D until sides BC and CD "look congruent". Now measure the base angles. What do you observe?