Download Kite and Quadrilateral with Proofs Lesson Plan

Kite and Quadrilateral with Proofs
Lesson Plan
By: Douglas A. Ruby
Class: Geometry
Date: 10/23/2003
Grades: 9/10
INSTRUCTIONAL OBJECTIVES:
This is a two part lesson designed to be given over a several day time span. At the end of the first
part of the lesson, the student will:
1. Have constructed a figure that meets the definition of a kite under all possible
translations.
2. Understand that there are multiple methods of construction of a figure meeting the
definition of a kite.
3. Discovered the relationships formed by triangles created by the diagonals of the kite.
4. Derive the area of the kite by observation of the areas of the embedded triangles
(induction).
5. Write a logical deductive (Euclidean) proof of the area of the kite based on the lengths of
its diagonals.
At the end of the second part of the lesson, the student will:
1.
2.
3.
4.
Be able to construct an arbitrary quadrilateral ABCD.
Construct midpoints.
Discover properties of interior quadrilateral formed by midpoints of ABCD.
Use Similar Triangle theorems to prove properties observed in prior step.
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Exploring Interior and Exterior angles of a Triangle with Geometer’s Sketchpad – Mr. Ruby
Relevant Massachusetts Curriculum Framework
10.G.1
10.G.2
10.G.3
10.G.4
10.G.5
10.G.6
10.G.7
Identify figures using properties of sides, angles, and diagonals. Identify the figures’
type(s) of symmetry.
Draw congruent and similar figures using a compass, straightedge, protractor, and other
tools such as computer software. Make conjectures about methods of construction.
Justify the conjectures by logical arguments.
Recognize and solve problems involving angles formed by transversals of coplanar
lines. Identify and determine the measure of central and inscribed angles and their
associated minor and major arcs. Recognize and solve problems associated with radii,
chords, and arcs within or on the same circle.
Apply congruence and similarity correspondences (e.g., ABC
XYZ) and properties
of the figures to find missing parts of geometric figures, and provide logical
justification.
Solve simple triangle problems using the triangle angle sum property and/or the
Pythagorean Theorem.
Using rectangular coordinates, calculate midpoints of segments, slopes of lines and
segments, and distances between two points, and apply the results to the solutions of
problems.
Draw the results, and interpret transformations on figures in the coordinate plane, e.g.,
translations, reflections, rotations, scale factors, and the results of successive
transformations. Apply transformations to the solutions of problems.
INTENDED AUDIENCE
This lesson is targeted at a 9/10th grade honors geometry audience. These students already know
how to use Windows 98/Me/XP based computers and are fluent with use of keyboard, mouse
movement, left and right mouse click, and normal conventions and terms such as <esc>, <tab>,
double-click, etc. Students have already taken Algebra (either in Middle school or high school)
and have already been introduced in prior classes to the notions and conventions of formal
proof, logic, and the basics of Theorems, postulates, definitions, and construction.
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Exploring Interior and Exterior angles of a Triangle with Geometer’s Sketchpad – Mr. Ruby
CLASS ACTIVITIES
Part 1A - Properties of a Kite
In our previous lessons, we learned how to use Geometer’s Sketch Pad Version 4.0.3. In this
lesson, we will build on that knowledge. Our purpose in this lesson is to create two
constructions, a kite, and an arbitrary quadrilateral. NBo new construction techniques will be
introduced. However, once done with our constructions, we will use traditional Euclidean
methodws of proof The construction that cannot easily be created using the traditional Euclidean
means of a compass and straightedge and then to observe and discover the properties of this
construction.
1. Create a new workspace and save it as A:kite.gsp
We begin by starting Geometer’s Sketchpad and using File>Save As to save the blank workspace
as A:kite.gsp.
2. Create Kite ABCD
A kite is defined as a quadrilateral that has two congruent sets of adjacent sides.
1) Which are the congruent adjacent sides? ____AB and AD are adjacent congruent sides as
well as CB and CD_____________________________________________________
There are a number of different ways to create a kite. You may decide to create a triangle and
find a way to cause that triangle to create a kite via a form of transformation (see Project 1).
2) What kind of transformation might produce a kite from a triangle? __A reflection over the
longest side would work_______________________________________________________
You may also decide to create a vertex of the kite and use another embedded construction figure
to guarantee that the two sets of adjacent sides of ABCD are congruent.
3) What kind of a figure produces this result? __We could use a circle and diameter. We would
then draw tow segments of equal length from a point on the diameter to the circle.
Do not proceed until you have answered these three questions
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Exploring Interior and Exterior angles of a Triangle with Geometer’s Sketchpad – Mr. Ruby
Method 1: Create a triangle ABC
B
C
A
Lets create triangle ABC as above. Once we do this, how might we get this triangle to look like
the figure below ____We would reflect triangle ABC over line AC._______________________
_____________________________________________________________________________
_____________________________________________________________________________
B
C
A
Once you have create this “kite like” figure, label the unlabeled vertex D. You may need to
create an intersection point to do this. You already have a diagonal AC, create diagonal BD and
label the intersection point E.
Continue to step 3 below (on the next page).
Method 2: Create two congruent adjacent sides.
Create two sides of the kite (AB and AD) that are congruent by
construction as in the following diagram. How might you do this?
____Draw a circle centered at A whose radius is eqal to AB and AD_
__________________________________________________________
__________________________________________________________
What happens if you place the Point Tool over point D and move it, are
the two sides still congruent? _____They should be________________
__________________________________________________________
__________________________________________________________
Do not turn the page until you have answered the questions on this page
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Exploring Interior and Exterior angles of a Triangle with Geometer’s Sketchpad – Mr. Ruby
There are multiple ways to construct segments AB and AD such that they are always congruent
to each other. This can be done by a transformation reflection of AB onto AD (suggested by
Method 1 above). It can also be done as in the following diagram formed by creating circle O
with point O as the center and point B on the circle.
In this case, what is segment BD? _____BD is a chord_________________________________
How would we construct line OE such that it always bisects segment BD? ____Construct a
perpendicular biisector____________________________________________________
What do we call line OE? ___OE is a diameter_______________________________________
If point A is any point on line OE, what do we know about segments AB and AD? ________
__They are congruent_________________________________________________________
Complete the construction of your kite by creating a point C such that segments BC and BD are
congruent. Make sure that the diagonals BD and AC have also been constructed.
3. Test your construction.
You now should have a construction that looks like this: (It may or may not have additional
lines, circles, or rays as construction lines). Make sure that the intersection of the diagonals is
labeled E and that the vertices of the
quadrilateral are labeled A, B, C, and
D as in the diagram to the left.
Measure the lengths of each of the
sides of the “kite” you created.
What are they? _Measurements will
vary_________________________
_____________________________
_____________________________
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Exploring Interior and Exterior angles of a Triangle with Geometer’s Sketchpad – Mr. Ruby
Write down the measurements of the adjacent sides of your kite.
AB = __AD___________
AD = __AB ___________
BC = __ DC ___________
DC = __BC_____________
Now place the Point Tool on one of the vertices of the kite. Move that vertex around by dragging
it. What happens to the four measurements? Write them down.
AB = __AD still___________
AD = __AB still___________
BC = __ DC still__________
DC = __BC still___________
Is your figure still a kite? If so, we want to test some other properties of your kite.
4. Embedded triangles.
We now want to observe and measure the properties of the triangles created by the diagonals of
the kite. What kind of triangles are AEB, AED, CEB, and CED? __Right triangles______
What measurement(s) would show this to be true? _ AEB, AED, CEB, and CED = 90o___
Are there any congruent triangles embedded within kite ABCD? If so, what are they? _________
__ AEB
AED, CEB,
CED_________________________________________________
Take measurements that “prove” that the triangles you mentioned are congruent. _____________
__Should measure angles and/or sides of all triangles_____________________________
Summarize the measurements of your kite as follows:
AB = __AD____________
BC = __DC____________
AC = __varies_________
AE = __ varies_________
CE = __ varies_________
m BAE = _ m DAE ___
m BCE = _ m DCE ___
m BEA = __90o_______
m DEA = __90o_______
AD = __AB____________
DC = __BC____________
BD = ___varies_________
BE = __DE = ½ BD_____
DE = __BE = ½ BD_____
m DAE = __ m BAE __
m DCE = __ m BCE __
m BEC = ___90o_______
m DEC = ___90o_______
5. Area of a Kite
We now want to measure the area of the kite ABCD. Given the measurements in section 4
above, do we have enough information to know the area of the kite? ___Yes__________
Conjecture: Write your own formula for the area of ABCD based on your own conjecture.
________AABCD = AC × BE__________________________________________________
Do not proceed until you have written your own formula.
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Exploring Interior and Exterior angles of a Triangle with Geometer’s Sketchpad – Mr. Ruby
We now need to construct interiors for the quadrilateral ABCD (the kite) as well as the congruent
sub-triangles created by the diagonals. We created interiors in prior lessons by selecting the
appropriate points and then using the Construct>Triangle Interior menu. You may also use the
Construct>Quadrilateral Interior menu. Measure the area of the congruent triangles and the kite
ABCD by selecting the appropriate interior of the construction and using Measure>Area.
m BAE = _ m DAE ___
m DAE = _ m BAE ____
m BCE = _ m DCE ___
m DCE = _ m BCE____
mABCD = _ m BAE+m DAE+m BCE+m DCE____
What relationship do you see? _____________________________________________________
Using the measurements taken in Section 4 for the segments and the formula you wrote in your
conjecture, use the Measure>Calculate tool to compute the area of ABCD. Is it equal to the area
measured by Geometers Sketchpad above? ___ Yes________________________________
6. Congratulations, you have gotten this far!
If you have gotten this far, you should have been able to construct a kite. Your construction
should meet the definition of a kite regardless of how it is stretched or moved. You should also
have a formula for the area of a kite based on the measurement of the diagonals of the kite.
If your formula is not based on the lengths of the diagonals, please write a new conjecture here
now: ____AABCD = ½(AC × BD)_________________________________________________
Now that you have created this construction, it is time to save it. Save your workspace to your
floppy disk using the File>Save menu.
Do not proceed until you have completed your construction, your measurements, and tested
your conjecture for the area of a kite relative to the lengths of its diagonals.
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Exploring Interior and Exterior angles of a Triangle with Geometer’s Sketchpad – Mr. Ruby
Part 1B – Formal Proof of the area of a kite
Using what you have discovered in Part 1A, we will proceed with a formal proof of the area of a
kite relative to the lengths of its diagonals. Please use the page provided below to proceed. Fill in
the blank parts of the proof. The first four steps have been given to you
Theorem:
The area of a kite is equal to on-half the product of the lengths of its diagonals.
Given:
Quadrilateral ABCD
AB AD and BC DC
Prove:
AABCD
AC BD
2
Statement
1. Quadrilateral ABCD
2. AB AD and BC DC
3. Diagonals AC and BD intersecting at E
4. AC AC
5. ABC ADC
6. AE AE
7.
BAC
DAC
8. BAE DAE
9.
BEA
DEA
10. m BEA = 90o
11. AABCD = A ABC + A
12. A ABC = A ADC
13. AABCD = 2×A ABC
14. BD = BE + DE
15. BE = DE
16. BD = 2×BE
17. BD/2 = BE
ADC
Reason
Given
Given
By construction
Reflexive
SSS
Reflexive
CPCTC
SAS
CPCTC
Congruent ’s that form a str. are right
’s.
Construction
Congruent ’s have equal area
Substitution
Segment Add. Postulate
CPCTC
Substitution from 14 & 15
Div Property of Equality
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Exploring Interior and Exterior angles of a Triangle with Geometer’s Sketchpad – Mr. Ruby
18. A ABC = (AC × BE)/2
19. A ABC = (AC × BD/2)/2
20. AABCD = 2×(AC × BD/2)/2
21. AABCD = (AC × BD)/2
Area of triangle (note BE is altitude)
Substitution from 17 & 18
Substitution from 13 & 19
Algebraic simplification of 20
Page 9
Part 2A – Midpoints of a Quadrilateral
In our previous lesson, we learned how to use Geometer’s Sketch Pad Version 4.0.3. In this
lesson, we will build on that knowledge. Our purpose in this lesson is to create a construction
that cannot easily be created using the traditional Euclidean means of a compass and straightedge
and then to observe and discover the properties of this construction.
1. Create a new workspace and save it as A:quad.gsp
We begin by starting Geometer’s Sketchpad and using File>Save As to save the blank workspace
as A:quad.gsp.
2. Create Quadrilateral ABCD
Create four points labeled A, B, C, and D. Construct segments connecting the points so that you
have an arbitrary quadrilateral labeled ABCD. This may look like this.
3. Create midpoints M, N, O, and P
Now, create a midpoint for each segment. The midpoint of segment AB should be labeled M.
The midpoint for segment BC should be labeled N. The midpoint of segment CD should be
labeled O. The midpoint of segment DA should be labeled P. Connect these four points with
constructed line segments.
Your quadrilateral should now look something like the following:
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Kite and Quadrilateral – Mr. Ruby
B
M
N
A
C
P
O
D
4. Observe the inner quadrilateral
Now we will do something new. Look carefully at quadrilateral MNOP. What kind of
quadrilateral does this appear to be? ___It may be a parallelogram_______________________
Try changing the appearance of quadrilateral ABCD by moving its vertices. Does the apparent
characteristic of MNOP: change is you change ABCD? __It shouldn’t change_____________
To highlight the inner quadrilateral MNOP, highlight the four segments using the point tool and
change the line width to “thick” using the Display>Line Width tool. Your quadrilateral should
look like the following now:
B
N
C
M
O
A
D
P
Write a conjecture regarding the properties of MNOP based on your observations:
Conjecture: _____The quadrilateral formed by connecting the midpoints of any arbitrary
quadrilateral will ALWAYS be a parallelogram. ________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
Do not proceed until you have written your conjecture.
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Kite and Quadrilateral – Mr. Ruby
5. Prove your conjecture
Let’s assume your conjecture has to do with opposite sides of quadrilateral MNOP being
parallel. How would you prove this within Geometers Sketchpad? _I could measure the opposite
angles of the “parallelogram” If the are equal, it is a poarallelogram. I could also measure the
lengths of the opposite sides. If they are equal, it is also a parallelogram.__________________
Provide the appropriate measurements to support your conjecture
m MNB = __varies_____
m AMP = __ varies_____
m DPO = ___ varies____
m CON= ___ varies____
m MPO = _ MNO ____
m NOP = _ PMN ____
MN = __OP___________
PM = __NO___________
m MNB = __ varies_____
m APM = __ varies_____
m DOP = ___ varies____
m CNO = ___ varies____
m MNO = _ MPO ____
m PMN = _ NOP ____
OP = ___MN__________
NO = ___PM__________
Which set of measurements provide evidence that your conjecture is true? ____The last 8
measurements are relevant_____________________________________________________
Is this sufficient for proof? __Yes it is, based on theorems proven earlier in class. We could also
measure slopes of the opposite sides to see that they are equal, thus parallel for direct analytic
proof by definition of a parallelogram.______________________________________________
Is there any benefit to drawing the diagonals BD and AC of quadrilateral ABCD? _maybe_____
Do this (draw the diagonals). Your diagram should now look like this:
How would we use AC and BD to prove MN || OP and NO || PM? __We would use Theorem
4.2.5 that states that the segment joining the midpoints of two side of a triangle is parallel to the
third side of the triangle and equal to ½ its length.____________________________________
Save your work now by doing a File>Save.
Do not proceed until you have completed the work above and saved your construction
Page 12
Kite and Quadrilateral – Mr. Ruby
Part 2B – Formal Proof of interior quadrilateral of a Square.
Using what you have discovered in Part 2A, we will proceed with a formal proof that the
quadrilateral formed by connecting the midpoints of a square is also a square. Please use the
page provided below to proceed. Fill in the blank parts of the proof. The first five steps have
been given to you.
Theorem:
The quadrilateral formed by connecting the midpoints of a square is also a square.
Given:
Quadrilateral ABCD is a square
M is the midpoint of AB, N is the midpoint of BC
O is the midpoint of CD, P is the midpoint of DA
Prove: MNOP is a square
1.
2.
3.
4.
5.
6.
7.
Statement
Quadrilateral ABCD is a square
M, N, O, and P are midpoints
AB BC CD DA
A
B
C
D
m A = m B = m C = m D = 90o
AC and BD are diagonals of ABCD
NO || MP
8. MN || OP
9. MN = OP= ½ AC
10. NO = MP= ½ BD
11. AC = BD
12. MN = NO = OP = PM
13. ABCD is a rhombus
14. m AMB=180o
15. AM = MB= ½ AB
16. BN = NC= ½ BC
17. CO = OD= ½ CD
18. DP = PA= ½ DA
Reason
Given
Given
Properties of a square
Properties of a square
Properties of a square
Construction
The segment joining the midpoints of two
side of a triangle is parallel to the third side
of the triangle and equal to ½ its length
“
“
“
Diagonals of a square are congruent
Substitution
Rhombus has 4 congruent/parallel sides
Definition of a straight angle
Bisector of a segment divides segment into 2
congruent segments
“
“
“
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Kite and Quadrilateral – Mr. Ruby
19. ½ AB = ½ BC = ½ CD = ½ AD
20. AM=MB=BN=NC=CO=OD=DP=PA
21. AMP is an isosceles right triangle
22. BMN is an isosceles right triangle
23. AMP
APM
BMN
BNM
24. m AMP + APM + m A = 180o
25. m AMP + AMP +90o = 180o
26. 2 × m AMP = 90o
27. m AMP = 45o
28. m BMN = 45o
29. m PMN + m AMP + m BMN = m AMB
30. m PMN = m AMB - m AMP - m BMN
31. m PMN = 180o – 45o – 45o = 90o
32. MNOP is a square
Multiplicative property of equality (from 3)
Substitution from 15-20
HL and two sides congruent
“
Angles opposite congruent sides of isosceles
triangle are congruent
Triangle sum Theorem
Substitution from 5 and 23
Subtraction property of identity
Division property of identity
Congruent angles have equal measure
Angle addition postulate
Subtraction property of identity
Substitution from 14, 27, and 28
A square is a rhombus with one right angle
Page 14
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