Project AMP
Dr. Antonio Quesada – Director, Project AMP
Diagonals of Special Quadrilaterals Lesson Summary: Students will investigate properties of the diagonals of parallelograms, rectangles, and rhombus. There are three activities in this lab. Activity one leads students to discover the properties of the diagonals of parallelograms. Activity two discusses rectangles and rhombuses are covered in activity three. Each lesson has an extension activity. Key Words: diagonals, parallelograms, rectangle, rhombus Background Knowledge: Students should be familiar with the Geometry software. Students should also understand the basic definitions of a parallelogram, rectangle, and a rhombus. Learning Objectives: Students will discover the properties of the diagonals of a parallelogram, a rectangle, and a rhombus. Materials: Geometry software Suggested Procedure: Split students into groups of two or three. Have students complete the worksheets. Each activity should take a class period. One suggestion is to have the groups do a different activity and then present findings to the entire class. Assessment: The completed worksheets can serve as assessment. One activity requires students to print out their software drawings.
Project AMP
Dr. Antonio Quesada – Director, Project AMP
Diagonals of Special Quadrilaterals Activity One: Diagonals of Parallelograms Team members’ names: _______________________________________________________________ File names: __________________________________________________________________________ Lab Goals: Students will discover properties of the diagonals of parallelograms. Investigate using Cabri: 1. Construct a point. 2. Label the point A. 3. Construct a horizontal line through point A. 4. Label this line l. 5. Construct a different line through point A and label it m. 6. Construct a point on line m and label it B. 7. Construct a line through point B and parallel to line l. 8. Construct a point on line l and label it point D. 9. Construct a line through point D and parallel to line m. 10. Draw a point at this new intersection and label it point C.
[Use point tool] [Use the label tool] [Use the perpendicular tool] [Use the label tool] [Use line tool] [point tool and label tool] [Use parallel line tool] [point tool and label tool] [Use parallel line tool]
[Intersection point tool]
Note: Your figure should be similar to the following construction.
11. Measure the sides and angles of figure ABCD. 12. Based on your knowledge from previous labs what type of quadrilateral is figure ABCD? _______________________________________________________________________________ _______________________________________________________________ 13. Construct diagonals AC and BD . [Segment tool] 14. Label the intersection of AC and BD point E [label tool] 15. Construct segments AE , BE , CE , and DE . [Use the segment tool] 16. Measure segments AE , BE , CE , and DE . [length and distance tool]
Project AMP
Dr. Antonio Quesada – Director, Project AMP
17. Drag these measurements to the side for future use.
[length and distance tool]
Note: Your figure should be similar to the following construction..
16. What relationship do you see between AE and EC ? How about BE and ED ? Where do the diagonal intersect? _____________________________________________________________________________________ ___________________________________________________________________ 17. Grab point B and move it. Grab point D and move it. Does the relationship between segments AE and EC and segments BE and ED remain the same for all parallelograms? _____________________________________________________________________________________ ___________________________________________________________________ We see that the diagonals of a parallelogram bisect each other. Will the following converse be true? Let’s find out! Converse: If the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram. Print out your figure and mark congruent parts as you work through the proof. Look at your construction. 18. Segments AD and EC are congruent and BE and ED are congruent. Do you think angles ∠AEB and ∠DEC are congruent? Why or why not? ______________________________________________________________________________ ______________________________________________________________ 19. Based on your observations above, what do you know about triangles ?AEB and ?DEC? ______________________________________________________________________________ ______________________________________________________________ 20. Use the same procedures as above, what do you know about ?AED and ?BEC? ______________________________________________________________________ 21. Therefore, what do we know about segments AB and CD ? How about segments BC and AD ? ______________________________________________________________________________ ______________________________________________________________
Project AMP
Dr. Antonio Quesada – Director, Project AMP
Diagonals of Special Quadrilaterals Activity Two: Diagonals of Rectangles Team members’ names: ___________________________________________________________ File names: __________________________________________________________________________ Lab Goals: Based on our previous labs, we know that if two pairs of opposite sides are congruent the figure is a parallelogram. Thus, figure ABCD is a parallelogram. Now, we know that our figure is a quadrilateral. How can we tell if it is a rectangle? Let’s find out in Activity Two. Use the following steps to create a Rectangle. 1. Construct a point A 2.
[Use the point tool]
Construct a line through point A and label it line l.
[Use the line tool]
3. Construct a line perpendicular to line l, through point A. Label this line m.
[Use the perpendicular tool]
4. Construct a point B on line m, but not point A
[Use the point tool]
5. Construct a line through point B and parallel to line l.
[Use the parallel line tool]
6. Construct a point D on line l, but not point A.
[Use the point tool]
7. Construct a line through point D parallel to line m.
[Use the parallel line]
8. Label the new point of intersection C
[Use the point tool]
9. Now construct segments AB , BC , CD , and DA .
[Use segment tool]
10. Hide all other lines except these segments.
[Use the hide tool]
Note: You should have a quadrilateral similar to the one below:
13. Construct diagonals AC and BD .
[Segment tool]
14. Measure segments AC and BD .
[length and distance tool]
Project AMP
Dr. Antonio Quesada – Director, Project AMP
15. Drag these measurements to the side for future use
[length and distance tool]
16. What relationship do you see between AC and BD ? _____________________________________________________________________________________ _____________________________________________________________________________________ 17. Grab point B and move. Grab point D and move it. Does these the relationship between segments AC and BD remain for all rectangles? _____________________________________________________________________________________ _____________________________________________________________________________________ This bring us to the following conclusion: Theorem: The diagonals of a rectangle are congruent.
Therefore, if you know a quadrilateral is a parallelogram, you can use the diagonals to determine if the quadrilateral is also a rectangle!
ExtensionActivity: There is a type of trapezoid with congruent diagonals. Determine the name of the figure, construct the figure and label the measurements of the diagonals to show this to be true. (hint – refer to your previous lab #7)
Project AMP
Dr. Antonio Quesada – Director, Project AMP
Diagonals of Special Quadrilaterals Activity Three: Diagonals of a Rhombus Team members’ names: _______________________________________________________________ File names: __________________________________________________________________________ Lab Goals: Now, let’s look at another figure. 1. Construct a point and label it point E.
[Use the point tool]
2. Draw two perpendicular lines through point E
[Use the perpendicular tool]
3. Label these line m and n.
[Use the label tool]
4. Construct a point on line m. Label this point A.
[Use the point tool]
Construct a point on line n and label it point B. Construct segment AB .
[Use segment tool]
Reflect segment AB about line n.
[Use the reflection tool]
Label this new endpoint C.
[Use the label tool]
Reflect the segment BC about line m.
[Use the reflection tool]
Label this new point D.
[Use the label tool]
Reflect this segment CD about line n.
[Use the reflect tool]
Hide lines m and n.
[Use the hide tool] Your figure should be similar to the following construction.
E
12. Measure the sides and angles of figure ABCD 13. What type of quadrilateral is figure ABCD? Justify your answer.
[length and distance tool]
Project AMP
Dr. Antonio Quesada – Director, Project AMP
______________________________________________________________________ 14. Construct diagonals AC and BD .
[Use the segment tool]
15. From the initial construction, what is the relationship between line m and n? _______________________________________________________________________________ _______________________________________________________________________________
This brings us to another theorem: Theorem #3: The diagonals of a _________________ are _____________________. 16. Measure ∠EAD and ∠EAB. Are these two angles congruent? Now measure ∠ECD and ∠ECB. Are they congruent? What can be said about segment AC . Complete the statement below:
Diagonal AC is the _____________ _____________ of ∠A and ∠C.
17. Use the same procedure from #16 to complete the following statement: Diagonal BD is the ______________ _______________ of ∠B and ∠D.
This brings us to a second theorem about the diagonals of a rhombus:
Theorem #4:Each diagonal of a rhombus ________________ two angles of the rhombus.
Extension Activity: Combining parts “a” and “b” of Lab #8…List four Theorems have we learned about a quadrilateral. Why do all of these apply to a square?
Project AMP
Dr. Antonio Quesada – Director, Project AMP
Journal Activity Diagonals of Special Quadrilaterals
1. What was your favorite thing about this activity?
2. What was the most challenging thing?
3. What did you gain the most confidence about through completing this lesson?
4. Where do you possibly see yourself using this knowledge in the future?
