Olive Oppong Transforming Mathematics with GSP 4 page 18 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 Olive Oppong Transforming Mathematics with GSP 4 page 19 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 M P S N M O Figure 2 2 a Q R Figure 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 W T U X Figure 2 3 Quadrilaterals Starting from 2 Free Points Olive Oppong Transforming Mathematics with GSP 4 page 20 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 Olive Oppong Transforming Mathematics with GSP 4 page 21 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 Olive Oppong Transforming Mathematics with GSP 4 page 22 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 M D N E C F P O Figure 2 5 A Cyclic Quadrilateral and a Quadrilateral with an Inscribed Circle Olive Oppong Transforming Mathematics with GSP 4 page 23 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