Olive & Oppong: Transforming Mathematics with GSP 4, page 1
Chapter 2: Exploring Quadrilaterals
In this chapter we extend our investigation of dynamically constructed figures in GSP to closed figures constructed from 4 line segments. These are known as quadrilaterals.
Activity 2.1: Starting with four free points
Open a new sketch and place 4 free points on the screen. Select all 4 points and choose Segment from the Construct menu. What kind of figure did you create? Depending on how you arranged your 4 points you could have very different looking figures from those of your classmates. Look around at other screens and compare the different figures. What is common to all of the figures? What is different? Move one or more of your points to create different kinds of quadrilaterals. Begin to think about how you might classify these different quadrilaterals. For instance, how might you classify the 3 figures shown in Figure 2.1?
Figure 2.1: Three Different Quadrilaterals
Activity 2.2: Starting with three free points
Open a new sketch and place 3 free points on the screen. Find a way to construct a fourth point that is dependent in some way on these 3 free points. Construct a quadrilateral using your 3 free points and your new constructed point. In what ways is you quadrilateral constrained? Which special quadrilaterals can you construct? Think about how you might classify all the quadrilaterals you can form with this construction. The 2 quadrilaterals in Figure 2.2 were constructed using different constructions but both started with 3 free points (M, N and O in Figure 2.2 a and P, Q and R in Figure 2.2 b).
Figure 2.2 aFigure 2.2 b
Activity 2.3: Starting with two free points
This time start with only 2 free points in your sketch and construct a quadrilateral. Remember to construct the other 2 points so that they are dependent upon the first 2 free points. Which special quadrilaterals can you construct? How does moving either of your 2 free points change your quadrilateral? Can you make more than one kind of special quadrilateral from your construction? The quadrilaterals in Figure 2.3 were constructed from the 2 labeled points using different construction methods.
Figure 2.3: Quadrilaterals Starting from 2 Free Points
Activity 2.4: Starting with the diagonals of a quadrilateral
Start with 2 arbitrary segments that intersect, construct the quadrilateral for which these segments are the diagonals – investigate the relations between the diagonals that create each of the special quadrilaterals you discovered in your explorations above.
In a new sketch, construct your diagonals so that they remain perpendicular to one another but have no other constraint (i.e. they do not have to be congruent; they do not have to bisect each other). Investigate the properties of all of the quadrilaterals you can form using your perpendicular diagonals. Can you find any “special” properties?
This time start with diagonals that remain congruent but have no other constraint. Investigate any special properties of these quadrilaterals.
Start with diagonals that bisect each other but have no other constraint. Investigate any special properties of these quadrilaterals.
Investigating midpoint quadrilaterals
Assignment 2.1: For each of the quadrilaterals formed by the special diagonal relations (perpendicular, congruent, or bisectors) construct the midpoint quadrilateral – that is the quadrilateral formed by connecting the midpoints of the 4 sides of the original quadrilateral (see Figure 2.4). In what ways are these midpoint quadrilaterals determined by the relation between the diagonals of the original quadrilateral?
Figure 2.4: A Quadrilateral Formed from Congruent Diagonals with Its Midpoint Quadrilateral
Construct the midpoint quadrilateral for a general quadrilateral (starting with 4 free points). Is there anything special about this midpoint quadrilateral?
Provide a rationale (an explanation or informal proof) for the special qualities of the midpoint quadrilaterals in each of the above cases. [Hint: Look for mid-segments of triangles.]
Classifying quadrilaterals
By now you should have several ways in which you could classify all of your quadrilaterals. Make a classification based on relations among the sides of a quadrilateral. Make another classification based on relations between the diagonals of a quadrilateral. Could you classify quadrilaterals based on properties of their mid-point quadrilaterals? Where do the “special” quadrilaterals fit in each of these different classifications? What other properties need to be considered to determine if a quadrilateral is a rectangle?
Quadrilaterals on a circle: another class of quadrilaterals
Start by drawing a circle using the Circle tool. Place 4 arbitrary points on the circle and use these 4 points to construct a quadrilateral. This figure is called a cyclic quadrilateral. Investigate the special properties of cyclic quadrilaterals. Make sure you move your points around the circle to make as many different cyclic quadrilaterals as possible. Find out which (if any) of your “special” quadrilaterals are cyclic. Test your conjectures by constructing the circle to pass through all 4 vertices of the quadrilateral. This circle is called the circumscribed circle of the quadrilateral.
Yet another class of quadrilaterals can be formed from those quadrilaterals for which a circle can be inscribed inside the quadrilateral so that each side of the quadrilateral is tangent to the circle. The easiest way to construct such quadrilaterals is to start with the inscribe circle, place four points on the circle and construct the four tangents at these points. Figure 2.5 shows both a cyclic quadrilateral (CDEF) and a quadrilateral (MNOP) with an inscribed circle. Find at least 2 properties unique to these special quadrilaterals. If a quadrilateral is cyclic can a circle also be inscribed inside it? If a quadrilateral has an inscribed circle is it also cyclic?
Figure 2.5: A Cyclic Quadrilateral and a Quadrilateral with an Inscribed Circle.
Investigating the Symmetry of Special Quadrilaterals
Look at the 2 quadrilaterals in Figure 2.2. Could you create one of them by constructing half of it and then reflecting that half about a mirror line? Which 3 points could you start with and which segment would become your mirror line? Any figure that could be constructed in this way is said to have “line symmetry.” This simply means that you can draw a mirror line somewhere across the figure in such a way that reflecting the figure about that line would not change the figure in any way (its reflected image would lie directly over the pre-image). Some figures may have more than one line of symmetry. This means that you could reflect the figure about two or more different lines and still not change the figure in any way. Do any of the quadrilaterals in any of the figures in this chapter have more than one line of symmetry? If so, construct all the lines of symmetry for those figures. Use the sketchpad Mark Mirror and Reflect actions to check your conjectures concerning lines of symmetry for the various quadrilaterals.
[GSP Notes: You can double click on any straight object to mark it as a mirror for reflection. Once a mirror has been marked, select the objects you want to reflect then go to the Transform menu and choose Reflect.]
Another form of symmetry is called rotational symmetry. A figure has rotational symmetry if you can rotate it about some point by less than 360 degrees (a full revolution) and its rotational image lies directly over the pre-image. In other words, the rotation does not change the appearance of the figure in any way. Do any of the quadrilaterals you have constructed so far have rotational symmetry? If so, determine the center of rotation and the smallest angle of rotation that will produce an identical image of the quadrilateral, lying directly over its pre-image. You can test your conjecture with GSP by constructing the center of rotation, marking it as a center of rotation by double clicking on the point, and then selecting all objects in your quadrilateral and choosing Rotate from the Transform menu. You will need to enter the angle of rotation (positive degrees for counter-clockwise rotation).
Can a quadrilateral have line symmetry without rotational symmetry? Can a quadrilateral have rotational symmetry without line symmetry? If so, provide examples. Make a classification of quadrilaterals based on symmetry.
Look for examples of both types of symmetry in nature. Does a human face have line symmetry? In Figure 2.6 the face on the right was constructed by reflecting the left half of the natural face on the left. Is the real face symmetric?
Figure 2.6: Face of a Maori Warrior
Constructing Similar Quadrilaterals
We found that for constructing similar triangles, all we had to do was to duplicate two of the angles of the triangle on a new segment. The intersection of the rays of the two angles would intersect in the third vertex of the similar triangle. We could then simply change the length of our new segment to form many triangles similar to the original triangle. Can we do the same for quadrilaterals? Do the angles of a quadrilateral fully determine its shape? To test this conjecture, create a quadrilateral using 4 free points. Draw a new segment. Starting with this segment duplicate the angles at each end of one side of your quadrilateral. This side will correspond to the new segment in your similar quadrilateral. For example, in Figure 2.7 segment EF corresponds to side AD of quadrilateral ABCD, angle FEF’ is congruent to angle DAB, and angle EFE’ is congruent to angle ADC.
Figure 2.7: Duplicating the Vertex Angles of One Side of a Quadrilateral on a New Segment.
[GSP Note: I used Rotation by Marked Angle to duplicate angles DAB and ADC on segment EF. First select points D, A and B (in that order) and choose Mark Angle from the Transform menu; then double click on point E to mark it as Center of Rotation. Select point F and choose Rotate from the Transform menu. F will be rotated about E to create point F’. Select the Ray tool, select points E and F’ in that order and construct the ray EF’. Use similar steps to rotate point E about point F by angle ADC and construct ray FE’.]
Unfortunately, the intersection of the two rays in Figure 2.7 (if it exists) does not give you another vertex of the similar quadrilateral. Regarding the quadrilateral ABCD as composed of 2 triangles could help you complete the construction of the similar quadrilateral and also help you “prove” that the shape of a quadrilateral is completely determined by its 4 angles (see Figure 2.8 for the completed construction.
Figure 2.8: Construction of a Similar Quadrilateral.
You can also use the Dilation tool to shrink or enlarge any figure you have constructed in GSP. You need to select a center of dilation by double clicking on any point in your sketch, then select the Dilation tool from the toolbar by holding the mouse button down over the Arrow tool. The Dilation tool is the furthest to the right of the arrow select tool options. Use this special selection tool to select the sides of your quadrilateral and then move these sides towards or away from the center of dilation. Your quadrilateral should change its size but not its shape.
Assignment 2.2: Investigate properties of similar quadrilaterals. Measure the ratios of the lengths of corresponding sides. Measure the areas of the similar quadrilaterals. How does the ratio of the areas relate to the ratio of corresponding side lengths? Will these properties generalize to similar polygons with more than 4 sides?
Constructing Congruent Quadrilaterals
With triangles, 3 sides completely determine a triangle (SSS congruency). Will 4 sides completely determine a quadrilateral? To test this conjecture, try using circle by center and radius to duplicate each side of a free quadrilateral starting with a point on a new line. For example, Figure 2.9 appears to show a possible construction for a duplicate of quadrilateral ABCD (FIJH).
Figure 2.9: Attempt to Duplicate a Quadrilateral by Duplicating Each Side Length.
Figure 2.10, however, clearly indicates that the 2 quadrilaterals are not necessarily congruent, even though they have the same side lengths!
Figure 2.10: Non-congruent Quadrilaterals with the Same Side Lengths.
Assignment 2.3: Determine the minimum set of properties to construct a quadrilateral congruent to a given quadrilateral. Use your minimum set to duplicate a quadrilateral in GSP.
Construction Problems for Special Quadrilaterals
Activity 2.5: Attempt at least one construction from each of the following sets, first with straight-edge and compass, then with GSP. Each given length (side, diagonal or altitude) should be a separate, free segment in GSP. Each given angle should be a set of 3 free points in GSP. Check that your construction works for all possible (or reasonable) cases by varying the free segments and free points that define your given elements. Create GSP custom tools for each of your constructions. The “givens” in each custom tool should correspond to the givens in each construction plus one or two free points on which to build your quadrilateral. Use comments in the script-view of your tool to let the user know which given GSP objects define which givens in the construction. Exchange your custom tools with other members of your class.
A. Construct a square, given the length of the diagonal
B. Construct a rhombus, given
1) one side and one angle
2) one angle and a diagonal
3) the altitude and one diagonal
C. Construct a parallelogram, given
1) one side, one angle, and one diagonal
2) two adjacent sides and an altitude
3) one angle, one side, and the altitude on that side
D. Construct an isosceles trapezoid, given
1) the bases and one angle
2) the median, altitude, and one of the bases
3) one base, the diagonal, and the angle included by them
Reflection. Look back at your constructions and ponder the following questions:
1)Which was easier – straight-edge and compass, or GSP for solving these construction problems?
2)What properties of the special quadrilaterals did you make use of in solving these problems?
3)Did manipulating the constructions in GSP help you to find errors in your construction?
4)Did some constructions only work for special relations among your givens (e.g. as long as the diagonal was longer than the side for the parallelogram)?
Investigating Golden Quadrilaterals
The investigations that follow will focus on four sided figures in which at least one pair of opposite sides are parallel. Among these there are four sided figures that have both pairs of their opposite sides in the same ratio. Special cases are the square, rhombus, rectangle, and parallelogram in which the opposite sides are in the ratio of one to one. Some of these figures have their adjacent sides in a special ratio. An interesting case of this is the Golden Rectangle with adjacent sides in the Golden Ratio.
The Golden Rectangle
The Golden Rectangle has the ratio of its adjacent sides as the Golden Ratio. The Golden Rectangle can be constructed using straight edge and compass. The construction is based on the fact that, the Golden Ratio can be expressed as , which is approximately 1.62. If a square of side 2 units is constructed, then the length from a vertex to the midpoint of the opposite side will be . Locate the midpoint of a side of a square and extend this side in either direction. Using the midpoint as center, draw an arc of radius equal to the length from the midpoint to an opposite vertex, to cut the extended side of the square. Constructing a perpendicular line at the point where the arc cuts the extended side of the square to meet the opposite extended side of the square can complete the construction. The resultant rectangle has one side of length 2, and the other 1+. The Golden Rectangle has a long history and many excellent references are available (see Huntley, 1970; Garland, 1987; and Vajda, 1989). It is a centerpiece in the Walt Disney classic film Donald in Mathmagic Land (1959).
Are there other quadrilaterals with adjacent sides or even opposite sides in the Golden Ratio? The Golden Rectangle when constructed with straight edge and compass is rigid. The dynamic features in Geometer's Sketchpad, however, permit us to transform the rectangle into a parallelogram, by moving three sides without moving the base (by changing the angle between base and side), and yet maintain the lengths and hence the ratio of its sides. The size of the whole figure can also be varied to show that the ratio does not depend on the size of the parallelogram. We will be able to generalize the notion of Golden Rectangle to the parallelogram.
The Golden Parallelogram
A dynamic Golden Parallelogram can be constructed by first constructing a Golden Rectangle CMSH (see figure below) and then drawing a circle with center C and passing through H. Next, place an object (point P) on the circle. Through P, construct PR parallel and equal to CM. As P is moved along the circle, the parallelogram CMRP will always have the ratio of its adjacent sides equal the Golden Ratio. We can use traditional methods or the dynamic features in the Geometer's Sketchpad to do this construction. A parallelogram with this property could be called a Golden Parallelogram. Due to the dynamic nature of the Geometer's Sketchpad, the distances and ratios shown besides the figure can be set to change accordingly on the computer monitor as the point P moves along the circle in the plane. None of the lengths or ratios of the sides, however, will change as long as P moves along the circle. But the lengths of the diagonals will change as P moves along the circle. In the figure CH=CP=MS=MR=2.79 cm. Also given are the corresponding measures for other sides and lengths, and the ratios of adjacent sides, and lengths of diagonals. You can investigate the range of these changes and associated ratios. Perhaps another Golden Ratio will turn up!