CK12 - Geometry

Supposethat is the longestside.Thenstatements2 and 3 aboveare true.

In orderto prove1. , constructthe perpendicularfrom point to on the opposite
side as follows:

Now we havetwo right trianglesand can drawthe followingconclusions:

Sincethe perpendicularsegmentis the shortestpath from a pointto a line (or segment),we have is

the shortestsegmentfrom to. Also, is the shortestsegmentfrom to. Therefore

and and by additionwe have

.

So,.

Example 1

Can you havea trianglewith sideshavinglengths?

Withouta drawingwe can still answerthis question—itis an impossiblesituation,we cannothavesucha
triangle.By the TriangleInequalityTheorem,we musthavethat the sum of lengthsof any two sidesof the

trianglemustbe greaterthan the lengthof the third side.In this case,we note that.

Example 2

Find the angleof smallestmeasurein the followingtriangle.