unit three plane and coordinate geometry 20 hours math … · any point on the perpendicular...

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UNIT THREE

PLANE AND

COORDINATE GEOMETRY

20 HOURS

Math 521B

Revised Dec 20, 00

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SCO: By the end of grade11, students will beexpected to:

E27 write proofs using various axiomatic systems

Elaborations - Instructional Strategies/SuggestionsGeometric Proofs (7.1)Challenge students to read and discuss p.385-389.Students will be expected to keep a Journal which includes definitions,theorems, postulates and corollaries in a separate notebook which maybe used in the unit exam.

Two new theorems are introduced here:

Perpendicular Bisector Theorem (proved in ex. 2 p.387)Any point on the perpendicular bisector of a line segment is equidistantfrom the endpoints of the segment.

Hypotenuse-Side Congruence Theorem(HS) (proved in ex. 5 p.388)

Note to Teachers:Students will generally not be able to do complete proofs on their ownin the time allotted in the course. The problems given are partially donefor the students with some blanks to be completed. Our expectationsmust be realistic in what most students will be able to do after only onesection on geometric proofs.

Students will need help in writing the congruency statements in thecorresponding order. They could be given a few exercises where thetriangles are congruent and they must write the congruency statement.Ex:Complete the following congruency statements:

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Worthwhile Tasks for Instruction and/or Assessment Suggested Resources

Geometric Proofs (7.1)Pencil/PaperComplete the following congruency statements:

Pencil/PaperComplete the following proof:

Geometric ProofsMath Power 11 p.389 #1-4,11,12, 15,25,27

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E14 investigate and prove theorems that are associated with circles, measurement and distance

Elaborations - Instructional Strategies/SuggestionsChord Properties (7.2)Initiate a discussion on the definition of a circle and the terms involvedwith a circle.A circle is a set of all points in a plane that are at a given distance(radius) from a given point in the plane (centre).

A Circle is named by its centre. The size is defined by its radius. Theterms diameter and circumference should be discussed as well.

Allow student groups time to read p. 396. Invite the groups to write statements that explain the terms “secant” and “chord”.

A secant is a line that cuts a circle at 2 points.A chord is that portion of a secant that is inside or on a circle.A diameter is a chord that contains the centre of a circle.Challenge students to construct, using a Reflect-View, a lineperpendicular to a given line through a point either on or not on thegiven line.Invite students to read and discuss the:Chord Perpendicular Bisector Theorem

a) The z bisector of a chord contains the centre of the circle

b) The z bisectors of 2 non-parallel chords intersect at the centre ofthe circle.

c) The z from the centre of a circle to a chord bisects the chord.

d) The line segment joining the centre of a circle and the midpoint of a

chord is z to the chord.

Note: (a) is proved in Ex. 1 p.397, (c) is proved in Ex. 2 p.397

Students should read and discuss the:Congruent Chords Theorem (proved in Ex. 4 p.399)If two chords are congruent, then they are equidistant from the centre ofa circle.

Converse of the above:If two chords are equidistant from the centre of a circle, then the chordsare congruent.

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Chord Properties (7.2)ManipulativeUse a Reflect-View to construct a line perpendicular to line“l” passing through point “A”.a) b)

Manipulative

Use a Reflect-View to construct a z bisector for thesegment:

Manipulative Use a Reflect-View to locate the centre of the given circle.

JournalGive clear and concise instructions that would allow anotherstudent to be able to locate the centre of any circle.

Chord PropertiesMath Power 11 p.400 #1-19,25,26, 33,36

Problem Solving StrategiesDescribe how you could find whereto drill holes for a bucket handleusing only a carpenter’s square.Explain your reasoning to the class.

Math Power p.395 Green #1-3

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E23 discover relationships within circles and use them to solve problems

Elaborations - Instructional Strategies/SuggestionsArc Length and Sector Area (7.6)Allow student groups to come to a consensus on the formula for thecircumference of a circle of radius r ( C = 2Br).As one travels in a circular path the distance travelled is 2Br units oflength and the number of degrees rotated is 3600. If half a completecircumference were travelled (a semi-circle) 1800 of rotation isinvolved. In general, a part of a complete circumference is called an arcand its length is called an arc length, “l”, defined by the formula.

l m r=FHGIKJ

0

03602π

Students should read and discuss p. 433 and do the first five inquireproblems on that page.Challenge the student groups to come up with the formula for the areaof a circle (A= 2Br) and to use the logic above to develop the formulafor the area of a sector of a circle.

A m r=FHGIKJ

0

02

360π

A sector is that part of a circle swept out by a rotation of m0. It lookslike a piece of pizza or pie.

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Arc Length and Sector Area (7.6)Group ActivityA lighthouse beacon rotates back and forth through a sectorangle of 1100. If the effective range of the light is 20 km, findthe sector area within which a mariner could see the light.

Group PresentationA rectangular pizza that is 40 cm by 60 cm has been cut into16 pieces. A circular pizza with a diameter of 40 cm has beencut into 8 pieces. Which pizza has larger slices?

Pencil/PaperMary had a piece of pie at the Big Stop in Aulac. The pie hasa diameter of 30 cm. If the piece has an arc length of 15 cm,find the sector angle and the sector area to the nearest tenth ofa cm.

Problem Solving Strategies1) Three towns are not collinear. Where should acommunications tower be built so that it is equidistant fromeach of the towns?

2) A carpenter wants to rip a board into three equal widthstrips. The board is 16 cm wide. How can it be done using a30 cm ruler?

Note to Teachers: See NCTM Principles and Standards 2000p. 345.

Arc Length and Sector AreaMath Power 11 p.435 # 1-14,33,34, 42

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E5 use inductive and deductive reasoning when observing patterns, developing properties and making conjectures

Elaborations - Instructional Strategies/SuggestionsAngles and Polygons (7.7)

Interior Angle Sum TheoremAllow student groups to develop notions regarding convex and concavepolygons.Students may find it helpful to read p.439 and do the explore andinquire activity on p.439-440 (See Worthwhile Tasks). This activityshould permit students to inductively obtain the Interior Angle SumTheorem: The sum of the interior angles of any polygon is 180(n!2)0.

Exterior Angle Sum TheoremThe sum of the exterior angles of any convex polygon is 3600.

Challenge student groups to draw a diagram of an exterior angle of apolygon. They should start by drawing an exterior angle of a triangle,then a quadrilateral, and so on.Have the students do the Exterior Angle Activity in the WorthwhileTasks so that they can make a conjecture about the sum of exteriorangles of convex polygons.

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Angles and Polygons (7.7)Pencil/PaperConstruct the polygons listed in the table. Divide eachpolygon into triangles by drawing diagonals from one vertexto all non-adjacent vertices. Complete the table below andwrite a conjecture about the sum of the interior angles of anypolygon.

Pencil/PaperMake a conjecture about the sum of the exterior angles of anyconvex polygon by completing the table below. The polygonshave been kept regular for simplicity’s sake.

DiscussionDeductively prove that the sum of the measures of theexterior angles of any convex polygon is 3600.

Angles and PolygonsMath Power 11 p.442 # 1-6,11,13-16, 20,22,23,26, 28,32-34,39

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E16 use the Cartesian Coordinate system to organize problems

Elaborations - Instructional Strategies/SuggestionsConnecting Coordinate and Plane Geometry (8.1)

Many proofs are accomplished easier using coordinate geometry. Ifgeometric figures are correctly placed on a Cartesian Coordinate Plane,then many conjectures about those figures can be deductively proved.Students should ensure their familiarity with the slope, midpoint anddistance formulae.

Invite students to read and discuss the Triangle Midsegment Theoremand its proof on p.458 in ex. 3(b).If a segment joins the midpoints of two sides of a triangle, then thesegment is parallel to and ½ the length of the third side.

Encourage students to research the following quadrilaterals and becomefamiliar with their properties. Students should enter this informationinto their Journals.

Parallelogram Rectangle Rhombus Square

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Connecting Coordinate and Plane Geometry (8.1)

Pencil/PaperDetermine the coordinates of the remaining points in thissquare:

Group PresentationUse coordinate geometry and the above figure to prove thatthe diagonals are perpendicular to each other and equal inlength.

Connecting Coordinate and PlaneGeometryMath power 11 p.460 # 1-3,5-7, 14-16,19-21 Worksheet at the end of unit.

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E16 use the Cartesian Coordinate system to organize problems

Elaborations - Instructional Strategies/SuggestionsCoordinate Matrices of geometric figures

Students must write the coordinates of a figure in matrix form.Ex:

The coordinate matrix for triangle ABC is:

Transformations using MatricesGeometric figures can be transformed four basic ways:Translation - a figure is slid a given distance without changing shape orsize. This can be done with matrix addition.Dilation - enlarges or shrinks a figure by a scale factor. This can bedone with scalar multiplication.Rotation - a figure turns through a given angle about a point called thecentre of rotation. This can be done with multiplication by a rotationalmatrix.Reflection - flips a figure over a given line (line of symmetry). This canbe done with multiplication by a reflection matrix.Geometric figures drawn on a coordinate plane can be defined using acoordinate matrix.

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Coordinate Matrices of geometric figuresPencil/PaperWrite the coordinate matrix of each figure in matrix:a) A(0,3), B(3,2), C(5,4), D(2,7).b) A(!2,!3), B(4,!1), C(6,4)

Pencil/PaperWrite the coordinate matrix of this figure:

Pencil/Paper Locate the figure defined by the matrix below on theCartesian Coordinate Plane.

Teachers can use the graph papertemplate on the school servers tocreate problems.

See Worksheets at the end of the unit

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E2 apply transformations when solving problems

E31 analyse properties of transformations in terms of mapping

Elaborations - Instructional Strategies/SuggestionsTranslations using MatricesTranslations of geometric figures can be done very easily with matrixaddition once the coordinates of the figure has been written in matrixform.

Ex. Translate ªABC 5 spaces to the left and 3 spaces down.

Solution

Note: The first row of the Translation matrix causes a horizontal shift.The second row of the matrix causes a vertical shift.

Computers define figures using coordinate matrices and move themaround the screen using the various transformation matrices talkedabout on this and the following pages.(Cartoons for instance or theanimations in movies like “The Matrix”.

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Translations using MatricesGroup ActivityTriangle ABC has coordinates (!1,1), (2,!3) and (4,!6).Construct the triangle on a coordinate plane and translate it 3spaces left and 4 spaces up. State the Translation Matrix thatwill achieve this translation and show the translation usingmatrices.

Pencil/PaperWhat is the Translation Matrix that describes the translationshown.

CommunicationWhat is the Translation Matrix that will cause a figure to beshifted 4 spaces to the right and 1 space up.

JournalExplain what each row of the Translation Matrixaccomplishes. With what operation is the Translation Matrixand the coordinate matrix of the figure combined?


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SCO: By the end ofgrade 11, students willbe expected to:

E2 apply transformations when solving problems


Elaborations - Instructional Strategies/SuggestionsDilations using MatricesInvite students to discuss their notions on the meaning of the term“Dilation”. It simply meaning enlarging or reducing a given figure by acertain factor. This factor is actually a scalar by which the coordinatematrix has been multiplied, thus creating a figure similar to the original.

If the scalar is: > 1, then the figure is enlarged < 1, then the figure is reduced

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Dilations using MatricesPencil/PaperWhat scalar has been used to reduce the figure as shown?

ActivityWhat scalar must triangle ABC, defined by A(0,1), B(4,!3)and C(!3,!1), be multiplied by to enlarge it three times?Show the original and enlarged figure on graph paper. Showthe enlargement process using matrices.

JournalExplain how a figure can be enlarged or reduced usingmatrices.PresentationUse matrices to show one enlargement and one reduction ofthe figure shown below. Explain your process to your group.


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E33 perform various transformations using multiplication of matrices

Elaborations - Instructional Strategies/SuggestionsReflections using Matrices

Challenge students to use a Reflect-View to reflect ªABC through the: < x-axis < y-axis < y =!x line < y = x line

The reflection matrices that are multiplied by the coordinate matrix of

ªABC are:

In general:through the x-axis: through the y-axis

through the y = ! x line: through the y = x line

These Reflection Matrices above will reflect any figure through thelines as described above.

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Reflections using MatricesManipulativeUse a Reflect-View to reflect triangle ABC through the x-axisand draw triangle ANBNCN.

Complete the matrix for triangle ANBNCN. Finally using onlythe numbers {!1,0 or 1} try to find the Reflection Matrix thatwhen multiplied by the matrix for triangle ABC yields thematrix for triangle ANBNCN.

Group ActivityUse the same procedure as above to find the ReflectionMatrix that will reflect triangle ABC above through:a) the y-axisb) the y = ! x linec) the y = x line


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E32 derive and apply the general rotational matrix

E33 perform various transformations using multiplication of matrices

Elaborations - Instructional Strategies/SuggestionsRotations using MatricesStudent groups can be challenged to rotate triangle ABC through 900,1800 and 2700 about the origin using a rotational matrix. The General Rotation Matrix is:

The 900 counterclockwise rotation matrix:



Challenge students to discuss the effects of multiplying the

a) for the 900 rotation matrix, the variables are reciprocated and theratios of x and y in the two x, y matrices have the opposite sign. (ie.they are negative reciprocals of each other; the corresponding sides are

z).

b) for the 1800 rotation matrix, the signs of both the x and y variablesare changed.

c) for the 2700 rotation matrix, the effect is the same as the 900 matrix.

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Rotations using MatricesGroup Activity

Tra ce theaxes and triangle ABC on a transparency. Stick a compassthrough the transparency and paper copy at the origin, thenrotate the transparency 900, 1800 and 2700 respectively todetermine the coordinates of the vertices for each of therotated triangles and write them in the matrices below. Next,using {!1, 0 or 1}, try to find the 900, 1800 and 2700 rotationalmatrices.

For a 900 rotation:

For a 1800 rotation:



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Worksheet for Connecting Coordinate and Plane Geometry

1. Determine the coordinates of the remaining points in each figure:

a) b)

c) d)

e) Group Presentation 2. Use Coordinate Geometry and the appropriate figure from question 1 to prove: a) the diagonals of a rectangle are congruent b) the diagonals of a square are perpendicular and equal in length c) the diagonals of an isosceles trapezoid are congruent d) the diagonals of a rhombus are perpendicular

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Coordinate Matrices of geometric figures

Pencil/Paper1. Write the Coordinate Matrix of each figure in matrix:a) A(0,3), B(3,2), C(5,4), D(2,7).b) A(!2,!3), B(4,!1), C(6,4)

Pencil/Paper2. Write the Coordinate Matrix of this figure:

Pencil/Paper 3. Locate the figure defined by the matrix below on the Cartesian Coordinate Plane.

Activity4. Write the Coordinate Matrix for the figure below.

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Translations using Matrices

Group Activity5. Triangle ABC has coordinates (!1,1), (2,!3) and (4,!6). Construct the triangle on a coordinate planeand translate it 3 spaces left and 4 spaces up. State the Translation Matrix that will achieve thistranslation and show the translation using matrices.

Pencil/Paper6. What is the Translation Matrix that describes the translation shown.

Communication7. What is the Translation Matrix that will cause a figure to be shifted 4 spaces to the right and 1 spaceup.

Journal8. Explain what each row of the Translation Matrix accomplishes. With what operation is the TranslationMatrix and the Coordinate Matrix of the figure combined?

Activity9. Construct a geometric figure on a coordinate plane and write its coordinate matrix. Translate the figureusing a Translation Matrix of your choosing. Finally write the Translation Matrix, the Coordinate Matrixand the Coordinate Matrix of the translated figure in an equation.

Group Project10. Make a 10 sheet(minimum) flip chart cartoon where, as the sheets are flipped, a geometric figureappears to move smoothly on the coordinate plane. On each sheet also show the Translation Matrix usedto translate the previous figure.

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Dilations using Matrices

Pencil/Paper11. Write the Coordinate Matrix for the dilation shown below, then draw the original and the dilatedfigure on a Cartesian coordinate plane.

Communication12. What scalar has been used to dilate the figure in the diagram below?

Activity13. What scalar must triangle ABC, defined by A(0,1), B(4,!3) and C(!3,!1), be multiplied by to enlargeit three times? Show the original and enlarged figure on graph paper. Show the enlargement processusing matrices.

Journal14. Explain how a figure can be enlarged or reduced using matrices.

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Presentation15. Use matrices to show one enlargement and one reduction of the figure shown below. Explain yourprocess to your group.

Pencil/Paper16. On a coordinate plane draw a figure, write its Coordinate Matrix, then dilate it so that it is:

a) enlarged by a factor of 2

b) reduced by a factor of 1/3.

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Reflections using Matrices

Manipulative17. Use a Reflect-View to reflect triangle ABC through the x-axis and draw triangle ANBNCN.

Complete the matrix for triangle ANBNCN. Finally using only the numbers {!1,0 or 1} try to find theReflection Matrix that when multiplied by the matrix for triangle ABC yields the matrix for triangleANBNCN.

Group Activity18. Use the same procedure as above to find the Reflection Matrix that will reflect triangle ABC abovethrough:a) the y-axisb) the y = ! x linec) the y = x line

Activity19. Use the appropriate Reflection Matrix to reflect triangle ABC with coordinates A(!2,!3), B(3,!1)andC(0,4) through:

a) the x-axisb) the y-axisc) the y = ! x line

Show neat diagrams illustrating these reflections. Show the original and reflected figures as solid anddashed figures respectively.

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Manipulative20. Use a Reflect-View to reflect the figure below through the y-axis and draw it on the coordinate plane:

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Rotations using Matrices

Group Activity21.

Trace the axes and triangle ABC on a transparency. Stick a compass through the transparency and papercopy at the origin, then rotate the transparency 900, 1800 and 2700 respectively to determine thecoordinates of the vertices for each of the rotated triangles and write them in the matrices below. Next,using {!1, 0 or 1}, try to find the 900, 1800 and 2700 rotational matrices.

For a 900 rotation:



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Activity22. Use the 90° Rotation Matrix to rotate the triangle below:

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Answers for Worksheet for Connecting Coordinate and Plane Geometry

1.a) D (0,a), C (a,a)b) D (0,b)c) C (a + b, c)d) C (a ! b, c)e) C (a + b, c)

2.

a) AC b a a b= − + − = +( ) ( )0 02 2 2 2

BD b a a b= − + − = +( ) ( )0 02 2 2 2

b) AC a a a= − + − =( ) ( )0 0 22 2 2

BD a a a= − + − =( ) ( )0 0 22 2 2

m a

a

m aa

AC

BD

= =

=−

= −

1

1

c) AC a b c

BD a b c

= − +

= − +

( )

( )

2 2

2 2

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d)

m ca b

m cb a

AC

BD

=+

=−

We must show that c

a bc

b a+⋅

−= −1

In this rhombus: AB BC

a a b a cignoring the square roots

a b c

=

− + − = + − + −

= +

( ) ( ) ( ) ( )0 0 0 02 2 2 2

2 2 2

Therefore c

a bc

b ac

b aa bb a+

⋅−

=−

= −−

= −2

2 2

2 2

2 2 1

unit three plane and coordinate geometry 20 hours math … · any point on the perpendicular...

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