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Chapter 9: Similarity Exploring Geometry 1999 Key Curriculum Press 189

Modeling a Pantograph Name(s):

A pantograph is a simple mechanical device that uses two pens to copyand enlarge or reduce drawings or maps. Thomas Jefferson made one,hoping he could use it to write more than one letter at a time. In thisactivity, youll do a very simple construction that does what a panto-graph does. Then, if youre brave and if you have enough time, youll

construct a model that more closely resembles a physical pantograph.

Sketch and Investigate

1. Construct fAB.

2. Construct point Con

fAB, beyond point B.

3. Select points Band C;then, in the Displaymenu, turn on TracePoint.

4. Drag point Bto write your name.

5. MeasureABandAC.

6. CalculateAB/AC.

7. Draw something with point B. Notice that point Cmoves on the rayso that the ratioAB/ACstays constant.

8. Move point Cto make a different ratio. Experiment drawing things

with point Busing different ratios.

Q 1 What does the ratio have to do with the traces of points Band C?

An actual, physical pantograph is constructed of rigid material, such asstrips of wood. These pieces dont stretch the way a dynamic Sketchpadray does. So an actual pantograph depends on linkages that make itflexible.

The following pages describe a construction that models a physicalpantograph.

A

B

C

Select points A and

B; then, in the

Measurem enu,

chooseDistance.

Repeat for AC .D ouble-click on a

m easurem ent to

activate the

calculator. C lick

once on a

m easurem ent to

enter it into a

calculation.

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190 Exploring Geometry 1999 Key Curriculum Press Chapter 9: Similarity

Modeling a Pantograph (continued)

Modeling an Actual Pantograph

9. In a new sketch, construct sAB.(This is not part of thepantograph, but its a controlsegment that will make parts ofyour pantograph both rigid andadjustable.)

10. Construct fCD.

11. Construct a circle with centerpoint Cand radiusAB.

12. Construct a circle with centerpoint Dand radiusAB.

13. Construct point Eat oneintersection of these circles. (If the circles dont intersect, drag pointDuntil they do.)

14. Construct fCE.

15. Construct sDE.

16. Hide the circles.

17. Construct sEF on fCE.

18. Construct a linethrough point Fparallel to sDE.

19. Construct a linethrough point Dparallel to fCE.

20. Construct point Gwhere these linesintersect.

21. Construct point Hat the intersection of dFG and fCD.

A B

C

D

E

Steps 915

H old the m ouse

button d ow n on the

Segmenttool to

show the Straight

Objectspalette.

D rag right to choose

the

Raytool.Select point C and

sAB ; then, in theConstructm enu,

chooseCircle By

Center+Radius.

A B

C

D

E

F

G

H

Steps 1621

Select point F

and sD E; then, inthe Construct

m enu, choose

Parallel Line.

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Chapter 9: Similarity Exploring Geometry 1999 Key Curriculum Press 191

22. Hide fCE, fCD,dFG,and dDG.

23. Construct sCE,sFG,sDG,andsGH. This is something likewhat a real pantographlooks like.

A B

C

D

E

F

G

H

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192 Exploring Geometry 1999 Key Curriculum Press Chapter 9: Similarity

Modeling a Pantograph (continued)

24. Construct sCD and sDH and make these segments dashed. Thesesegments wouldnt appear on a real pantograph, but they can helpyou see how a pantograph works.

25. Drag point Dto observe how the pantograph behaves. Note that it

falls apart if you drag point Dtoo far from point C. You can extendits range by lengthening sAB.

26. Turn on TracePointsfor points Dand H.

27. Drag point Dtotrace out yourname.

28. Move point F,thendrag point Dto seehow the location ofpoint Faffects thetrace of point H.

Q 2 How would you locate point Fso that the trace of point Hwas twiceas large as the trace of point D? Use similar triangles to explain why.

Explore More

1. Build an actual pantograph out of old rulers, small bolts, andwing nuts.

After you draw sC D ,chooseDisplay:

Line Weight:

Dashed.

A B

C

D

E F

G

H

This m ay take

several tries.

Experim ent w ith

different starting

places for point D . If

necessary, m akesAB

longer and m ove

point F farther from

point E.