• Aucun résultat trouvé

AB = BE = EF = F G = GC = 1

N/A
N/A
Protected

Academic year: 2022

Partager "AB = BE = EF = F G = GC = 1"

Copied!
2
0
0

Texte intégral

(1)

Homework #7

In the gure below, the angles

ABC [

and

BCD \

are right and, in units,

AB = BE = EF = F G = GC = 1

,

BC = 4

and

CD = 2

. The segments linkingthe verties

A

and

D

to every point onthe segment

[ BC ]

have been drawn.

B C

D

A

E F G

Part A – Static experiments with Geogebra

1. Reprodue the pitureusing Geogebra.

2. (a) UseGeogebratoomputethe lengths

AE

,

AF

,

AG

and

AC

.Giveineahase anapproximationto 3DP.

(b) Do the same for the lengths

DB

,

DE

,

DF

and

DG

.

() Two of the eightlengths you've omputed are equal. Explain why.

3. (a) UseGeogebratoompute thesums

AB + BD

,

AE + ED

,

AF + F D

,

AG + GD

and

AC + CD

to 3DP.

(b) Whihof these 5 sums is the lowest? Do yound this surprising?

Part B – Dynamic experiments with Geogebra

1. Do anew gure with Geogebra,with only the points

A

,

B

,

C

,

D

and the segments

[ AB ]

,

[ BC ]

and

[ CD ]

2. Dene the lengths

m = AM

and

n = M D

and then their sum

d = m + n

.

3. Ask Geogebra toshowthe values of the numbers to5 DP, and move

M

along

[ BC ]

to nd the position where

d

isminimal.Zoominif youneed more preision.

4. Write downthe valueof

BM

suh that

d

isminimal.This value islose toasimple fration, whihone?

Part C – Using functions

In this part,we name

x

the distane

BM

and

d ( x )

the sum ofdistanes

AM + M D

. 1. Write the lengths

M C

,

AM

, and

M D

asfuntions of

x

.

2. Givethe expression of

d ( x )

asa funtion of

x

.

3. Drawthe graph ofthis funtion with Geogebra orwith your alulator.Print itor,

if youan't, draw iton your paper.

(2)

4. Find graphially,and with

5

DP, the minimum of the funtion

d

and the values of

x

for whih this minimum isreahed.

5. Explain how the result of the previous question onrms the onlusion of part A.

Part D – Some properties of the minimal point

Let

E

be the point onthe segment

[ BC ]

suhthat

BE = 1 3

− − − − − →

BC

.

1. Do a new gure with Geogebra and plae preisely the point

E

. Print itor opy it on your paper atthe end of the exerise.

2. What an yousay about the diretions ofthe vetors

− − − − − →

BE

and

− − − − − →

CE

? Compute their norms (lengths) as frations.

3. What an you dedue about the sum

2 BE + CE

?

4. Use the denition of

E

and Chasles' relationtoprove againthe equality youfound in the previous question.

5. Use Geogebra to buildthe point

A

, symmetriof

A

aroundthe point

B

. 6. Drawthe line

( A D )

. What doyounotie?

7. Whatisthepositionofapoint

M

onthesegment

[ BC ]

suhthatthesum

A M + M D

is minimal? Dedue an explanation of the property you notied in the previous

question.

Références

Documents relatifs

[r]

To handle the case of partial differential operators the abstract theory o f symmetric relations on a Hilbert space may be used.. The abstract theory was first

Exprimer les sommes suivantes en fonction d’un vecteur défini par deux points de la

The “Live@Knowledge 5.0” computer program will be presented to all those people who shared their data and knowledge to build knowledge needed to solve historical problems.. It is

Vecteurs, cours pour la classe de seconde.. Vecteurs, cours

For measurements of the HBAR noise in extended conditions, a dedicated phase noise setup inspired from [10], using carrier suppression techniques (to reduce the sustaining amplifier

Banica introduced the notions of dephased and undephased de- fects for matrices F G , and showed that they give upper bounds for the tangent spaces at F G to the real algebraic

The considered scheme of the algorithm for GI (exhaustive search on all bijections that is reduced by equlity checking of some invariant characteristics for the input graphs) may