> :<9!` 4bjbj\\ 4 >>4((((D.lllllllloqqqqqq$hPlllllll000l
llo0lo00rTl`pZ^(v
co0q`ll0lllll&
lllllll$
D
Objective
To find the shortest
distance from a point
to a line to a point on the
same side of the line.
The Shortest Path Problem
Introduction:
A PE teacher designs a relay race where one student stands 16 feet from a wall and another student stands 12 feet from the same wall, the students are 45 feet apart along the wall. On the teachers command of GO, the first student must touch the wall then touch the other student in the fastest time, competing with other teams of students. Josh and Jake have been selected to go first, knowing that to have the fastest time, Josh must run the path that produces the shortest distance, he runs to the middle of the segment along the wall that is between the them. Will this path be the shortest? If not what path would be the shortest?
Construction & Exploration
Part I: Construct a scaled model.
Draw a line to represent the wall. Move the two points that
define the line to the side of the screen so that they are out of
the way.
Place two points on the same side of the line. Label the
points Ja, and Jo.
Construct a perpendicular from point Ja to the line. Find
the intersection point and label it A. Repeat for point Jo,
label the intersection point B.
Measure the distance JaA, JoB, and AB. Use scaling
techniques to make sure the distances are to scale with
the problem as stated above.
Place a point on the line between points A and B. Label
the point C.
Draw segments from Ja to point C, and Jo to point C.
Part II: Data collection.
Measure the length of the segment JaC and JoC.
Use Calculate to find the sum of JaC and JoC.
Use the alpha key to move point C along the line to find
the shortest distance from Ja to point C to Jo.
Questions and Conjectures
For the shortest path found above, does point C appear to be the midpoint of segment AB?
Reflect point Ja over the line. Connect the reflection of point Ja with point Jo. Make a conjecture about the shortest path.
Why does the intersection point of the line and the segment JaJo produce the point on the wall where Josh must touch the wall?
!"89UVmn 2 L N p ָָָsdXLhs-CJOJQJaJhCJOJQJaJh'hGM/CJOJQJaJhQvbCJOJQJaJh'h)LCJOJQJaJh'CJOJQJaJh'hCJOJQJaJh'hvCJOJQJaJh'hvCJOJQJaJh'h'CJOJQJaJh'CJOJQJaJh'ht}CJOJQJaJh'hvCJOJQJaJ
"9Vmn45WX/
&Fgd~cgd)Lhh^h`hgd~c
&Fgd)L^`gd=^v4 4W{./012kl⸬❬~occTht;>ht;>CJOJQJaJht;>CJOJQJaJh'h~cCJOJQJaJh)LCJOJQJaJ%jh^Oh~cCJOJQJUaJh'hGM/CJOJQJaJh~cCJOJQJaJhCJOJQJaJhhvCJOJQJaJh'hvCJOJQJaJh'h)LCJOJQJaJh'hs-CJOJQJaJ/12l
<
Y
Z
"
&Fgd=^vhh^h`hgdQvbgdx^`gdt;>hh^h`hgdt;>
&Fgd)L^`gd~c
;
<
@
"#RST4ŶԶԶteVh'hHCJOJQJaJh=^vhHCJOJQJaJh=^vhxCJOJQJaJhxCJOJQJaJh=^vCJOJQJaJhQvbCJOJQJaJh'hxCJOJQJaJh'hGM/CJOJQJaJh'ht;>CJOJQJaJht;>CJOJQJaJhGM/CJOJQJaJ%jh^Oht;>CJOJQJUaJ"#ST45234gdH
&FgdHgdx`gd=^v
&F
gd=^v
&Fgd=^vgd=^v21h:pt;>/ =!"#$%Dd
<90
#Ab,>y~;kh;8i&wDT n>y~;kh;8i&wPNG
IHDR{pgAMA|QIDATH
pB_% >U^X?lIM('°<.F8$ᒅq)<zr:ힴZCBBBNZȶw4wd<-іd!-1.+,Fn#W#r&Bo1
.s8ϜP$IENDB`Dd
KI0
#Ab@$Ot75=ћTT n$Ot75=ћTPNG
IHDR{pgAMA|QIDATH
p렯Z:ث`]7Ve5a0(cq=PwGbr r*d 9;rtKҥB.
&=}>*%LIUl(
ZZ>9!sk5v]Dn7P*N,B9Pj0H;"=](bIENDB`H@HNormal CJOJQJ_HaJmH sH tH DA@DDefault Paragraph FontRi@RTable Normal4
l4a(k@(No List4
"9Vmn45WX/12l<YZ"#ST452360000000000000000 0000 0000 000000 0000 000 0000 00 00 00000 00 00 000
"9Vmn45WX12l60000000000000000 000000H0000000000 4 /"4
466
""99VV02"#RT366v1:\YJu.%k^`ko(.^`o(.
pLp^p`LhH.
@@^@`hH.
^`hH.
L^`LhH.
^`hH.
^`hH.
PLP^P`LhH.eke^e`ko(.
^`hH.
L ^ `LhH.
^`hH.
xx^x`hH.
HLH^H`LhH.
^`hH.
^`hH.
L^`LhH.^`o(.
^`hH.
pLp^p`LhH.
@@^@`hH.
^`hH.
L^`LhH.
^`hH.
^`hH.
PLP^P`LhH.eke^e`ko(.
^`hH.
L ^ `LhH.
^`hH.
xx^x`hH.
HLH^H`LhH.
^`hH.
^`hH.
L^`LhH.:u.%Y z?< r :( S1's-/GM/t;>Qvb~c=^vt}d/vxH)L@l4@@UnknownGz Times New Roman5Symbol3&z Arial7Georgia?5 z Courier New"h˻^F899#r4d002QHX)?v2 ObjectiveRogerRogerOh+'0p
,8
DPX`hObjectiveRogerNormalRoger4Microsoft Office Word@P@r @B9՜.+,0hp|
0\
ObjectiveTitle
!"#$%&'(*+,-./02345678;Root Entry F04p^=Data
1TableWordDocument4 SummaryInformation()DocumentSummaryInformation81CompObjq
FMicrosoft Office Word Document
MSWordDocWord.Document.89q