Season 1 • Episode AP06 • Using Geogebra to solve a problem

Season 1 • Episode AP06 • Using Geogebra to solve a problem

⁰

Using Geogebra to solve a problem

Season 1

Episode AP06 Time frame 1 period

Prerequisites :

^None.

Objectives :

•

^{Usinggeogebra toonjueture and solvea problem.}

Materials :

•

^{Computer room.}

•

^{Exerise sheets.}

1 – Part A 20 mins

Using geogebra and working by pairs, Students have to draw a gure and to onjeture

the maximalvalue of anarea.

2 – Part B 20 mins

Studentshavetondout the funtioninvolved inthe problemandthentodrawit(using

geogebra) in order tond the maximal value of the area.

3 – Part C 20 mins

In this part, Students have to onrm their previous answers, this time using omputa-

tions.

Using Geogebra to solve a problem ^{Episode AP06}

Document ICT

In the gure below,

ABC

^{is a} right-angled isoseles trianglein

A

^{suh that}

BC = 8

^m.

M N P Q

^{is a retangle insribed in}

△ ABC

^{. The length}

BM

^{is variable, we denote it}

x

^,

in m.

The aim of this session isto nd the position of the point

M

^{that maximizesthe area of}

the retangle

M N P Q

^.

B N

A

P

C

M Q

x

Part A – Using Geogebra

1. Use Geogebra to draw the gure. Don't forget to display the area of the retangle

on the gure.

2. Movethepoint

M

^onthesegment

BC

^{andonjeturethemaximalvalueofthearea}

of

M N P Q

^.

Part B – Using the graph of a function

1. What is the minimal value for

x

^{? Explain quikly your answer.}

2. What is the maximal value for

x

^?

3. What are the measures of the angles of

△ ABC

^{? of}

△ BM N

^{? Dedue the nature}

of the triangle

BM N

^{then the length of}

M N

^{as afuntion of}

x

^.

Season 1 • Episode AP06 • Using Geogebra to solve a problem

²

4. Let

f

^{bethe funtion thatmaps any adequatevalueof}

x

^tothe orrespondingarea of

M N P Q

^{. Prove that}

f (x) = 8x − 2x ²

5. UseGeogebratodrawthe graphof

f

^{andgivethemaximalvalue of}

f

^andthevalue

of

x

^{for whih it isreahed.}

6. Dedue the maximalarea of

M N P Q

^{and thepositionof thepoint}

M

^forwhihit's

reahed.

Part C – Using computation

In this part, let's admit that the area of

M N P Q

^{is given by the expression}

8x − 2x ²

^,

where

x

^{represents the length}

BM

^.

1. What is the sign ofthe expression

− 2(x − 2) ²

^?

2. Expand the expression

−2( x − 2) ²

^.

3. Usingthetwopreviousquestions,dedue themaximalvalue ofthe areaof

M N P Q

^.

In the gure below,

ABC

^{is a} right-angled isoseles trianglein

A

^{suh that}

BC = 8

^m.

M N P Q

^{is a retangle insribed in}

△ ABC

^{. The length}

BM

^{is variable, we denote it}

x

^,

in m.

The aim of this session isto nd the position of the point

M

^{that maximizesthe area of}

the retangle

M N P Q

^.

B N

A

P

C

M Q

x

Part A – Using Geogebra

1. Use Geogebra to draw the gure. Don't forget to display the area of the retangle

on the gure.

2. Movethepoint

M

^onthesegment

BC

^{andonjeturethemaximalvalueofthearea}

of

M N P Q

^.

The maximum area for

M N P Q

^{seems tobe}

8

^m

²

^.

Part B – Using the graph of a function

1. What is the minimal value for

x

^{? Explain quikly your answer.}

The minimalvalue for

x

^is

0

^beause

x

^{represents adistane.}

2. What is the maximal value for

x

^?

The maximal value for

x

^is

4

^beause

M

^{an't be on the other side of the}

midpoints.

3. What are the measures of the angles of

△ ABC

^{? of}

△ BM N

^{? Dedue the nature}

of the triangle

BM N

^{then the length of}

M N

^{as afuntion of}

x

^.

△ ABC

^isaright-angledisoselestrianglein

A

^so

∠BAC = 90 ^o

^and

∠ABC =

∠BCA = 45 ^o

^.

In

△ BM N

^,

∠BM N = 90 ^o

^beause

M N P Q

^{is a retangle and}

∠M BN =

∠CBA = 45 ^o

^{. So}

△ BM N

^{is a}right-angledisoseles trianglein

B

^.

We dedue that

M N = BC = x

^.

Season 1 • Episode AP06 • Using Geogebra to solve a problem

⁴

4. Let

f

^{bethe funtion thatmaps any adequatevalueof}

x

^tothe orrespondingarea of

M N P Q

^{. Prove that}

f (x) = 8x − 2x ² f ( x ) = M N × M Q = x × (8 − 2 x ) = 8 x − 2 x ²

^.

5. UseGeogebratodrawthe graphof

f

^{andgivethemaximalvalue of}

f

^andthevalue

of

x

^{for whih it isreahed.}

Aording to the graph, the maximal value of

f

^{seems to be}

8

^{reahed for}

x = 2

^.

6. Dedue the maximalarea of

M N P Q

^{and thepositionof thepoint}

M

^forwhihit's

reahed.

Thefuntion represents thearea sothe maximalvalue ofthe area seemstobe

8

^m

²

^{reahed for}

BM = 2

^m.

Part C – Using computation

In this part, let's admit that the area of

M N P Q

^{is given by the expression}

8x − 2x ²

^,

where

x

^{represents the length}

BM

^.

1. What is the sign ofthe expression

−2( x − 2) ²

^?

( x − 2) ²

^{is asquaredthus positive.As wemultiplyitby}

−2

^{whihis negative,}

we dedue that

− 2(x − 2) ²

^{is negative.}

2. Expand the expression

− 2(x − 2) ²

^.

− 2(x − 2) ² = − 2(x ² − 4x + 4) = − 2x ² + 8x − 8

^.

3. Usingthetwopreviousquestions,dedue themaximalvalue ofthe areaof

M N P Q

^.

Aording tothe rst question

− 2(x − 2) ²

^{is negativeso}

− 2(x − 2) ² 6 0

^.

Aordingtotheseondquestion,weget

− 2x ² +8x − 8 6 0

^thus

− 2x ² +8x 6 8

^.

The lastinequationmeans that

f ( x )

^{an't be greater than}

8

^.

In the gure below,

ABC

^{is a} right-angled isoseles trianglein

A

^{suh that}

BC = 8

^m.

M N P Q

^{is a retangle insribed in}

△ ABC

^{. The length}

BM

^{is variable, we denote it}

x

^,

in m.

The aim of this session isto nd the position of the point

M

^{that maximizesthe area of}

the retangle

M N P Q

^.

B N

A

P

C

M Q

x

Part A – Using Geogebra

1. Use Geogebra to draw the gure. Don't forget to display the area of the retangle

on the gure.

2. Movethepoint

M

^onthesegment

BC

^{andonjeturethemaximalvalueofthearea}

of

M N P Q

^.

Part B – Using the graph of a function

1. What is the minimal value for

x

^{? Explain quikly your answer.}

2. What is the maximal value for

x

^?

3. What are the measures of the angles of

△ ABC

^{? of}

△ BM N

^{? Dedue the nature}

of the triangle

BM N

^{then the length of}

M N

^{as afuntion of}

x

^.

4. Let

f

^{bethe funtion thatmaps any adequatevalueof}

x

^tothe orrespondingarea of

M N P Q

^{. Prove that}

f ( x ) = 8 x − 2 x ²

5. UseGeogebratodrawthe graphof

f

^{andgivethemaximalvalue of}

f

^andthevalue

of

x

^{for whih it isreahed.}

6. Dedue the maximalarea of

M N P Q

^{and thepositionof thepoint}

M

^forwhihit's

reahed.

Part C – Using computation

In this part, let's admit that the area of

M N P Q

^{is given by the expression}

8 x − 2 x ²

^,

where

x

^{represents the length}

BM

^.

1. What is the sign ofthe expression

− 2(x − 2) ²

^?

2. Expand the expression

−2( x − 2) ²

^.

3. Usingthe twopreviousquestions, dedue the maximalvalue of the area of

M N P Q

and the position ofthe point

M

^{for whihit's reahed.}