Homework #7
In the gure below, the angles
ABC [
andBCD \
are right and, in units,AB = BE = EF = F G = GC = 1
,BC = 4
andCD = 2
. The segments linkingthe vertiesA
andD
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
andAC
.Giveineahase anapproximationto 3DP.(b) Do the same for the lengths
DB
,DE
,DF
andDG
.() 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
andn = M D
and then their sumd = 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 thatd
isminimal.This value islose toasimple fration, whihone?Part C – Using functions
In this part,we name
x
the distaneBM
andd ( x )
the sum ofdistanesAM + M D
. 1. Write the lengthsM C
,AM
, andM D
asfuntions ofx
.2. Givethe expression of
d ( x )
asa funtion ofx
.3. Drawthe graph ofthis funtion with Geogebra orwith your alulator.Print itor,
if youan't, draw iton your paper.
4. Find graphially,and with
5
DP, the minimum of the funtiond
and the values ofx
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 ′
, symmetriofA
aroundthe pointB
. 6. Drawthe line( A ′ D )
. What doyounotie?7. Whatisthepositionofapoint
M
onthesegment[ BC ]
suhthatthesumA ′ M + M D
is minimal? Dedue an explanation of the property you notied in the previous
question.