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

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## Using Geogebra to solve a problem

None.

### •

Usinggeogebra toonjueture and solvea problem.

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.

(2)

## Using Geogebra to solve a problem EpisodeAP06

### Document ICT

In the gure below,

### ABC

is a right-angled isoseles trianglein

suh that

m.

### M N P Q

is a retangle insribed in

. 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

.

B N

A

P

C

M Q

### 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

onthesegment

### BC

andonjeturethemaximalvalueofthearea

of

.

### Part B – Using the graph of a function

1. What is the minimal value for

### x

2. What is the maximal value for

### x

?

3. What are the measures of the angles of

? of

### △ BM N

? Dedue the nature

of the triangle

### BM N

then the length of

as afuntion of

.

(3)

2

4. Let

### x

tothe orrespondingarea of

. Prove that

### f (x) = 8x − 2x 2

5. UseGeogebratodrawthe graphof

### f

andgivethemaximalvalue of

andthevalue

of

### x

for whih it isreahed.

6. Dedue the maximalarea of

### M N P Q

and thepositionof thepoint

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

,

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

.

(4)

In the gure below,

### ABC

is a right-angled isoseles trianglein

suh that

m.

### M N P Q

is a retangle insribed in

. 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

.

B N

A

P

C

M Q

### 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

onthesegment

### BC

andonjeturethemaximalvalueofthearea

of

### M N P Q

.

The maximum area for

seems tobe

m

.

### Part B – Using the graph of a function

1. What is the minimal value for

### x

The minimalvalue for

is

beause

### x

2. What is the maximal value for

### x

?

The maximal value for

is

beause

### M

an't be on the other side of the

midpoints.

3. What are the measures of the angles of

? of

### △ BM N

? Dedue the nature

of the triangle

### BM N

then the length of

as afuntion of

.

### △ ABC

isaright-angledisoselestrianglein

so

and

.

In

,

beause

### M N P Q

is a retangle and

. So

### △ BM N

is aright-angledisoseles trianglein

.

We dedue that

.

(5)

4

4. Let

### x

tothe orrespondingarea of

. 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

andthevalue

of

### x

for whih it isreahed.

Aording to the graph, the maximal value of

seems to be

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

m

reahed for

m.

### Part C – Using computation

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

### M N P Q

is given by the expression

,

where

### x

represents the length

### BM

.

1. What is the sign ofthe expression

?

### ( x − 2) 2

is asquaredthus positive.As wemultiplyitby

whihis negative,

we dedue that

### − 2(x − 2) 2

is negative.

2. Expand the expression

.

### − 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

is negativeso

### − 2(x − 2) 2 6 0

.

Aordingtotheseondquestion,weget

thus

### − 2x 2 +8x 6 8

.

The lastinequationmeans that

### f ( x )

an't be greater than

### 8

.

(6)

In the gure below,

### ABC

is a right-angled isoseles trianglein

suh that

m.

### M N P Q

is a retangle insribed in

. 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

.

B N

A

P

C

M Q

### 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

onthesegment

### BC

andonjeturethemaximalvalueofthearea

of

.

### Part B – Using the graph of a function

1. What is the minimal value for

### x

2. What is the maximal value for

### x

?

3. What are the measures of the angles of

? of

### △ BM N

? Dedue the nature

of the triangle

### BM N

then the length of

as afuntion of

.

4. Let

### x

tothe orrespondingarea of

. Prove that

### f ( x ) = 8 x − 2 x 2

5. UseGeogebratodrawthe graphof

### f

andgivethemaximalvalue of

andthevalue

of

### x

for whih it isreahed.

6. Dedue the maximalarea of

### M N P Q

and thepositionof thepoint

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

,

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.

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