CK12 - Geometry

• Identifyacutetrianglesfrom side measures.


• Identifyobtusetrianglesfrom side measures.


• Classifytrianglesin a numberof differentways.

Converseof the PythagoreanTheorem

In the last lesson,you learnedaboutthe PythagoreanTheoremand how it can be used.As you recall,it
statesthat the sum of the squaresof the legs of any right trianglewill equalthe squareof the hypotenuse.

If the lengthsof the legs are labeled and , and the hypotenuseis , then we get the familiarequation:

TheConverseof the PythagoreanTheoremis also true. That is, if the lengthsof threesidesof a triangle

makethe equation true, then they representthe sidesof a right triangle.

With this converse,you can use the PythagoreanTheoremto provethat a triangleis a right triangle,even
if you do not knowany of the triangle’s anglemeasurements.

Example 1

Doesthe trianglebelowcontaina right angle?

This triangledoesnot haveany right anglemarksor measuredangles,so you cannotassumeyou know
whetherthe triangleis acute,right,or obtusejust by lookingat it. Take a momentto analyzethe side lengths

and see how they are related.Two of the sides and are relativelyclosein length.The third side

is abouthalf the lengthof the two longersides.

To see if the trianglemightbe right,try substitutingthe side lengthsinto the PythagoreanTheoremto see

if they makesthe equationtrue.The hypotenuseis alwaysthe longestside,so shouldbe substituted

for. The othertwo valuescan represent and and the orderis not important.

Sinceboth sidesof the equationare equal,thesevaluessatisfythe PythagoreanTheorem.Therefore,the
triangledescribedin the problemis a right triangle.

In summary, example1 showshow you can use the converseof the PythagoreanTheorem.The Pythagorean

Theoremstatesthat in a right trianglewith legs and , and hypotenuse ,. The converse