of the PythagoreanTheoremstatesthat if , then the triangleis a right triangle.
IdentifyingAcuteTriangles
Usingthe converseof the PythagoreanTheorem,you can identifywhethertrianglescontaina right angle
or not. However, if a triangledoesnot containa right angle,you can still learnmoreaboutthe triangleitself
by usingthe formulafrom PythagoreanTheorem.If the sum of the squaresof the two shortersidesof a tri-
angleisgreaterthan the squareof the longestside,the triangleisacute(all anglesare less than ). In
symbols,if then the triangleis acute.
Identifyingthe "shorter"and "longest"sidesmay seemambiguousif sideshavethe samelength,but in this
case any orderingof equallengthsidesleadsto the sameresult.For example,an equilateraltrianglealways
satisfies and so is acute.
Example 2
Is the trianglebelowacuteor right?The two shortersidesof the triangleare and. The longestside of the triangleis. First find the
sum of the squaresof the two shorterlegs.
The sum of the squaresof the two shorterlegs is Comparethis to the squareof the longestside,The squareof the longestside is Since , this triangleis not a right
triangle.Comparethe two valuesto identifywhichis greater.
The sum of the squareof the shortersidesis greaterthan the squareof the longestside.Therefore,this is
an acutetriangle.
IdentifyingObtuseTriangles
As you haveprobablyfiguredout, you can provea triangleisobtuse(has one anglelargerthan ) by
usinga similarmethod.Find the sum of the squaresof the two shortersidesin a triangle.If this valueis less
than the squareof the longestside,the triangleis obtuse.In symbols,if , then the triangleis