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Math 3 Geogebra Discovery - Quadrilaterals Decemeber 5, 2014

Discovery of the Properties of Quadrilaterals

As you work through this activity you’ll discover various properties of important quadrilaterals.But before we go there, we should get some vocabulary down.

Definition of a Polygon - A polygon is a two-dimensional, closed, rectilinear figure.

So what does it mean to be “closed”?Think about a fence that encloses animals. If the fence is closed then there is an inside and

an outside and the animals can’t get out. But if the fence is open then there is no boundary andthe animals can get out and run amoc! (Amoc is an awesome word, means to get into all kinds oftrouble.)

inside

outside

Open Figure Closed Figure

What does it mean to be ”rectilinear”?You have heard the root “rect–” before. Think of rectangle. Rectilinear means that the figure is

made up of only segments, not curves (remember that there is no such thing as a curvy segment!)

RectinlinearMade of only segment

Non-RectilinearContains at least one curve

1. Are all triangles polygons? Why?

2. Are circles polygons? Why?

Quadrilateral

We have already spent a lot of time with triangles (Remember: ”tri” three, angle, a polygonwith three angles.) We discussed that triangles have six characteristics that we can discuss:three sides, three angles. It gets a lot more complicated with four angles!

Definition of a Quadrilateral - A polygon with four sides.

(No idea why we don’t call them quadrangles...)

We need a few more definitions before continuing:

Convex Polygon - A polygon such that if you connect any two non-consecutive vertices theconnecting segment will be inside the polygon. Or - A polygon such that every interior angleis less than 180 degrees.

Concave Polygon - A polygon that is not convex.

Convex Concave

226.7◦

We will only be discussing convex quadrilaterals.

Vocabulary in a convex Quadrilateral:

Opposite Sides

Opposite Angles

Opposite in this case isthe same as non-consecutive.

DiagonalsSegments connectingnon-consecutive vertices.

Parallelograms

Your first objective today is to discover the properties of a parallelogram.

Definition of a Parallelogram - A parallelogram is a quadrilateral with opposite sides parallel.

By constructing a parallelogram on GeoGebra you will get to discover the properties in allparallelograms.

Directions:

1. Open a new Geometry window in GeoGebra and save it as “Parallelogram Properties -Your Name.”

2. To make one side of your parallelogram, use the segment tool and draw a segment somewherein the middle bottom of the window. Name the endpoints A and B Then draw anothersegment by clicking on the endpoint A of the first segment and then somewhere else. Itshould be at some angle to the first segment, it doesn’t matter what kind of angle. Namethe endpoint of the new segment D. You have just created two sides of your parallelogram.

3. Now we know from the definition that the other two sides of the parallelogram must beparallel to the first two. Under the perpendicular line tool, you’ll find the parallel line tool.To use the tool, click first on one of the segments you created, let’s click on AB. Then clickon point D. A line should appear that is parallel to AB. Now create the other side of theparallelogram by creating a line parallel to AD.

4. The two lines you just created will have an intersection point. Using the intersection tool,create and label a point C at the intersection.

5. Now use the polygon tool (not the “regular polygon” tool) and connect A to B to C to D.You should now have a parlalellogram named ABCD on your screen.

Side Note: When naming quadrilaterals they are always named in either a clockwise orcounterclockwise direction, in order. We would not name the parallelogram above ACBD,but we could name it DCBA.

6. By right clicking (Control click on a PC, command click on a mac) on one of the lines,you can get into “object properties.” Look in the left column and hide the original twosegments and the two lines. All you should see on your screen is the parallelogram withlabeled vertices.

The Drag Test: Whenever we are using GeoGebra to prove something it’s really importantthat the figure doesn’t depend on the placement of the segments or objects. So, when yougrab any vertex or any side of the parallelogram you should be able to drag the figureand make it shorter or fatter or skinnier or taller. Despite all this, it should still be aparallelogram!

Drag the parallelogram to make it shorter, fatter, skinnier, taller and check thealgebra view. Do the opposite sides remain parallel? How do you know?

7. In the algebra view you should be able to find the side lengths of the parallelogram. If youcannot find them, go ahead and measure them.

8. Then use the angle measurement tool to measure each of the parallelogram’s interior angles.

Fill in the table below, make a differently shaped parallelogram between eachentry by dragging a vertex or side.

AB BC AD DC1st Paralellogram

2nd Parallelogram

3rd Parallelogram

m∠A m∠B m∠C m∠D1st Paralellogram

2nd Parallelogram

3rd Parallelogram

9. Now using the segment tool, draw the diagonals of the parallelogram. In a convex quadri-lateral, there will always be two diagonals. They connect opposite vertices. The diagonalswill intersect, using the intersection tool, create and name this point M .

MA MC MD MB1st Paralellogram

2nd Parallelogram

3rd Parallelogram

Based on your data, finish the following properties of a parallelogram.

1. In a parallelogram, opposite sides are parallel (definition).

2. In a parallelogram, opposite sides are...

3. In a parallelogram, opposite angles are...

4. In a parallelogram, consecutive angles are...

5. The diagonals of a parallelogram, ...

Homework

IMPORTANT! IMPORTANT! You may not use the properties during theseproofs! You are proving the properties in these proofs.

1. Given: CAMP is a parallelogram.

Prove: CA ∼= MPCP ∼= AM

P M

C A

Statements Reasons


Prove: ∠C ∼= ∠M∠P ∼= ∠A

P M

C A

Statements Reasons


Prove: AP bisects CMCM bisects AP

P M

C A

Statements Reasons

Rectangles, Rhombuses and SquaresToday you will construct a rhombus, a rectangle and a square. Then using prior knowledge you

will determine the properties of each shape.

Definition of a Rectangle - A quadrilateral with four right angles.

Definition of a Rhombus - A quadrilateral with all four sides congruent.

Definition of a Square - A quadrilateral with all four sides congruent and four right angles.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Creating a Rectangle:

Open a new GeoGebra window and save it as “Rectangle - Your Name.” You may either attemptto create a rectangle on your own, or follow the instructions below. If you do it on your own, makesure you read through the instructions afterwards for the naming conventions and come back to theworksheet for the Drag Test and Measurement.

1. First draw a segment somewhere in the middle-bottom of the screen. Name the endpoints Eand C.

2. Now use the perpendicular tool to construct a line perpendicular to EC at E.

3. Name a point on the new line R. Using the perpendicular tool, construct a line through Rperpendicular to RE. You should now have two of the right angles of your rectangle.

4. To create the last two right angles, make a line perpendicular to EC through C. Name theintersection point of the fourth angle T . You should now have a rectangle named RECT . Usethe polygon tool to connect R, E, C and T in order to make the rectangle. Hide the lines andsegments not part of the rectangle.

5. Finally, draw in the diagonals and name the intersection point M .

Measurements and the Drag Test

When you created your rectangle you should be able to drag any vertex or side and still have arectangle, maybe just a skinnier, taller, shorter or fatter one.

What must remain constant every time you move a vertex or side in order for theshape to remain a rectangle?

Measure anything needed to help determine if the dragged rectangle remains a rectangle.

Creating a Rhombus

Open a new GeoGebra window and save it as “Rhombus - Your Name.” Remember that arhombus is a quadrilateral with all four sides congruent. In this case we also want it to not be asquare. You may either attempt to create a rectangle on your own, or follow the instructions below.If you do it on your own, make sure you read through the instructions afterwards for the namingconventions and come back to the worksheet for the Drag Test and Measurement.

1. Construct a segment in the middle of the page, it should be 2-3 inches on your screen. Namethe endpoints A and

2. Now use the perpendicular bisector tool (found under the perpendicular tool) and create aperpendicular bisector of the segment in part 1.

3. Now that we have a perpendicular bisector, any point on it will be equidistant to A and C. Sothis means we can easily get two sides congruent. Pick a point on the perpendicular bisectorand construct a point. Name it B.

4. Construct segments AB and BC. These two segments are congruent.

5. Now the trick is to make the other two sides congruent to the first two! GeoGebra won’t let youconstruct a segment of a certain length, so we need another method. We are going to use the“reflect about a line” tool. It’s next to the button that says ABC and has a red point on oneside of a diagonal line and a blue point on the other side. Select this tool, then click on pointB and then on AC. This let’s us reflect B over the segment AC. Name the reflected point D.

6. Construct DA and DC.

7. Use the polygon tool and click on A, B, C and D in order. Hide everything other than thepolygon.


When you created your rhombus you should be able to drag any vertex or side and still have arhombus, maybe just a skinnier, taller, shorter or fatter one.

What must remain constant every time you move a vertex or side in order for theshape to remain a rhombus?

Measure anything needed to help determine if the dragged rhombus remains a rhombus.

Creating a Square

Open a new GeoGebra window and save it as “Square - Your Name.” Remember that a Squareis a quadrilateral with all four sides congruent and four right angles. Notice that this makes a squareboth a rhombus and a rectangle! You may either attempt to create a square on your own, or followthe instructions below. If you do it on your own, make sure you read through the instructionsafterwards for the naming conventions and come back to the worksheet for the Drag Test andMeasurement.

1. Now there is a “regular polygon” tool that would quickly make a square. But I want you toactually see how to construct one without that. There are many ways to do this, this is justone of them.

2. Start by constructing a segment and name the endpoints A and B.

3. Construct two lines perpendicular to AB, one through point A and one through point B. Thiscreates one side and two right angles.

4. Now use the “Circle with center through point” tool. Create a circle with center A and radiusAB (do this by first clicking on point A and then clicking on point B). Then create a circlewith center B and radius AB.

5. Where the circles intersect the perpendicular lines, create points, C goes on the same line as Band D goes on the same line as A.

Why were the circles helpful? What do we know about AB, BC and AD?

6. Use the polygon tool to connect A to B to C to D in order. Hide all objects that are not thesquare.


When you created your square you should be able to drag any vertex or side and still have asquare, maybe just a bigger or littler one.

What must remain constant every time you move a vertex or side in order for theshape to remain a square?

Measure anything needed to help determine if the dragged square remains a square.

To determine the properties of the three shapes you have created work to fill out the chartbelow. Measure anything you need to determine your answers. Make sure that you only answerafter checking a few variations on each shape.

For instance, is a square a rectangle? Is a rectangle a square? There are things thatmight be true, but only sometimes

Property Parallelogram Rectangle Rhombus Square

Are the oppositesides parallel?

Are the oppositesides congruent?

Are the oppositeangles congruent?

Are all fourangles congruent?

Are all four sidescongruent?

Are the diagonalscongruent?

Are the diagonalsperpendicular?

Do the diagonalsbisect each other?

Do the diagonals bisectthe opposite angles?

Homework

IMPORTANT! IMPORTANT! You may not use the properties during these proofs!You are proving the properties in these proofs. You may only use the given definitionsof each shape.

1. Given: STAR is a square.

Prove:←→TR SA←→SA TR

S R

T A

Y

Statements Reasons

2. Given: ROCK is a rhombus.

Prove:←→OK bisects ∠ROC and ∠RKC←→RC bisects ∠KRO and ∠KCO

R

C

S

O

K

Statements Reasons

3. Given: BLUE is a Rectangle.

Prove: BU ∼= LE

B

E

L

U

S

Statements Reasons

Kites and Trapezoids

Last but not least, the black sheep of the quadrilateral family. These two shapes don’t fit withthe others quite like the last four. Throughout this activity, see if you can figure out why!

Definition of a Kite - A quadrilateral with two disjoint sets of sides congruent.

(Disjoint means that the sets don’t share anything, there is no overlap.)

Definition of a Trapezoid - A quadrilateral with one and only one set of sides parallel.

Definition of an Isosceles Trapezoid - A quadrilateral with one and only one set of sides paralleland the non-parallel sides congruent.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Kites

So, you’ve seen kites before, remember the PBC? We were calling it a kite just to help rememberwhat it looked like, but it turns out there is actually a shape called a Kite. Interestingly the“fly it in the air” kite is actually named after a bird called a kite!

To help us communicate about Kites without having to overexplain ourselves and to help usremain specific, here is some helpful vocabulary. Fill in the corresponding boxes on the diagram.

• The base diagonal is the diagonal that acts as a base of the two isosceles triangles formedby the congruent sides.

• The base vertices are the vertices of the Kite that are intersected by the base diagonal.

• The medial diagonal is the diagonal that divides the kite exactly in half (the two trianglesit creates are congruent).

• The medial vertices are the vertices of the Kite that are intersected by the medial diagonal.

I

E

K TM

Trapezoids

So the history behind the name Trapezoid is interesting. First of all, only in American andCanadian English is it called a Trapezoid, in all other English speaking countries it is calleda Trapezium. Trapezoid comes from Greek for “little table.” The first time it was called aTrapezoid was sometime between 412-458 CE by Marinus Proclus in his commentary on thefirst Euclid’s Elements.

There is a little extra vocabulary with trapezoids. Given the information below, fill in the boxeson the diagram.

• The bases of a trapezoid are parallel.

• The lower base angles share the lower base as one of their sides.

• The upper base angles share the upper base as one of their sides. (Assuming the trapezoidis horizontal, if it’s vertical you just pick a top and bottom).

• The non-parallel sides are called the legs.

T P

RA

Creating a Kite

Open a new window in GeoGebra and using your knowledge of the EQT, create a kite. Savethe file as “Kite - Your Name.”

Write two properties (not including the definition) of a Kite by finding the appropriate measureson your diagram. Make sure to move it around and see if it always works!

1.

2.

Creating a Trapezoid

Open a new window in GeoGebra and save it as “Trapezoid - Your Name.” You can try makingone on your own, or follow the instructions below.

1. Draw a segment on the screen and label the endpoints (left one) A and (right one) B. Thenusing the parallel tool, create a line parallel to AB.

2. Create two points, anywhere, on the parallel line. Name them (left one) D and (right one)C. Then create segments AD and CB.

3. Use the polygon tool to connect A to B to C to D in that order. Hide all other objects.

A general Trapezoid does not have any special properties beyond the definition. However, itis worth noting that Same-Side Interior Angles are Supplementary in a trapezoid (due to theparallel lines).

Creating an Isosceles Trapezoid

Constructing an Isosceles Trapezoid is quite a bit trickier than a normal trapezoid because thelegs have to be congruent. However, I’ll give you a hint before sending you off: If you have anisosceles triangle and you chop off the top of the triangle so that the cut is parallel to the base,you’ll be left with an isosceles trapezoid.

Isosceles Trapezoid

Write two properties (not including the definition) of anIsosceles Trapezoid by finding the appropriate measureson your diagram. Make sure to move it around and see if italways works!

1.

2.

Homework

Homework is a little different tonight. Instead of proving the properties (mostly because youalready know how to prove them!), you’re going to combine your knowledge of the past fewdays and solidify your understanding.

1. For each picture below, annotate everything you know about the shape. Include both thedefinition and the properties.

Rhombus

R

C

S

O

K

Parallelogram

T

O

S

N

Y

Rectangle

B

E

L

U

S

Trapezoid (bases BE and RA)

R A

EB

Kite (medial diagonal: PR)

O

K

P RY

Isosceles Trapezoid (bases SP and OR)

OR

S P

K

2. Complete the diagram below by placing the proper term in its appropriate position. Foran extra challenge, figure out where “Kite” would go and draw in the region.

Trapezoid, Rectangle, Square, Quadrilateral, Rhombus, Parallelogram, Isosceles Trapezoid

3. So you know how I said you could make an Isosceles Trapezoid by chopping the top off ofan isosceles triangle? Now you’re going to prove it! Because while I am highly reliable,proofs are better!

Remember: The definition of an isosceles trapezoid is that two sides are parallel and theother two sides are congruent, so you’ll have to prove both of those things!

Given: 4ABC is an isosceles triangle with base CBDE||CB

Prove: DEBC is an isosceles trapezoid.

A

BC

D E

Statements Reasons

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