HistoryNote
In ancienttimes,mathematicianswereinterestedin bisectingand trisectinganglesand segments.Bisection
was no problem.Theywereable to use basicgeometryto bisectanglesand segments.
But whatabouttrisection—dividingan angleor segmentinto exactlythreecongruentparts?This was a real
challenge!In fact, ancientGreekgeometersprovedthat ananglecannotbe trisectedusingonly compass
and straightedge.
With the LinedNotebookPaperCorollary, though,we havean easyway to trisecta givensegment.
Example 5
Trisectthe segmentbelow.Drawequallyspacedhorizontallines like linednotebookpaper. Thenplacethe segmentonto the horizontal
lines so that its endpointsare on two horizontallines that are threespacesapart.
- slantedsegmentis samelengthas segmentabovepicture
- endpointsare on the horizontalsegmentsshown
- slantedsegmentis dividedinto threecongruentparts
The horizontallines now trisectthe segment.We coulduse the samemethodto dividea segmentinto any
requirednumberof congruentsmallersegments.
LessonSummary
In this lessonwe beganwith the basicfactsaboutsimilartriangles—thedefinitionand the SSS and SAS
properties.Thenwe built on thoseto createnumerousproportionalrelationships.First we examinedpropor-
tionalsidesin triangles,then we extendedthat conceptto dividingsegmentsinto proportionalparts.We fi-
nalizedthoseideaswith a notebookpaperpropertythat gaveus a way to dividea segmentinto any given
numberof equalparts.
Pointsto Consider
Earlierin this bookyou studiedcongruencetransformations.Theseare transformationsin whichthe image
is congruentto the originalfigure.You foundthat translations(slides),rotations(turns),and reflections(flips)
are all congruencetransformations.In the next lessonwe’ll studysimilaritytransformations—transformations
in whichthe imageissimilarto the originalfigure.We’ll focusondilations. Theseare figuresthat we zoom