CK12 - Geometry

Afterlookingat the table,we mightmakethis conjecture:

The valueof is for any wholenumbervalueof.

However, if we try othervaluesof , suchas , we have

Obviously, our conjectureis wrong.For this conjecture, is calledacounterexample, meaningthat
this valuemakesthe conjecturefalse.(Of course,it was a prettypoor conjectureto beginwith!)

Example 8

Ramonastudiedpositiveevennumbers.She brokesomepositiveevennumbersdownas follows:

Whatconjecturemightbe suggestedby Ramona’s results?

Ramonamadethis conjecture:

“Everypositiveevennumberis the sum of two differentpositiveodd numbers.”

Is Ramona’s conjecturecorrect?Can you find a counterexampleto the conjecture?

The conjectureis not correct.A counterexampleis. The only way to makea sum of two odd numbers

that is equalto is: , whichis not the sum ofdifferentodd numbers.

Example 9

Arturis makingfiguresfor a graphicart project.He drewpolygonsand someof their diagonals.

Basedon theseexamples,Arturmadethis conjecture:

If a convexpolygonhas sides,then thereare diagonalsfrom any givenvertexof the polygon.

Is Artur’s conjecturecorrect?Can you find a counterexampleto the conjecture?

The conjectureappearsto be correct.If Arturdrawsotherpolygons,in everycasehe will be able to draw

diagonalsif the polygonhas sides.