UsingSimilarRightTriangles
LearningObjectives
- Identifysimilartrianglesinscribedin a largertriangle.
- Evaluatethe geometricmeanof variousobjects.
- Identifythe lengthof an altitudeusingthe geometricmeanof a separatedhypotenuse.
- Identifythe lengthof a leg usingthe geometricmeanof a separatedhypotenuse.
Introduction
In this lesson,you will studyfiguresinscribed,or drawnwithin,existingtriangles.One of the mostimportant
typesof linesdrawnwithina right triangleis calledanaltitude.Recallthat the altitudeof a triangleis the
perpendiculardistancefrom one vertexto the oppositeside.By definitioneachleg of a right triangleis an
altitude.We can find one morealtitudein a right triangleby addingan auxiliaryline segmentthat connects
the vertexof the right anglewith the hypotenuse,forminga new right angle.
You may recallthis is the figurethat we usedto provethe PythagoreanTheorem.In right triangle
above,the segment is an altitude.It beginsat angle , whichis a right angle,and it is perpendicular
to the hypotenuse. In the resultingfigure,we havethreeright triangles,and all of themare similar.
InscribedSimilarTriangles
You may recallthat if two objectsare similar, correspondinganglesare congruentand their sidesare pro-
portionalin length.In otherwords,similarfiguresare the sameshape,but differentsizes.To provethat two
trianglesare similar, it is sufficientto provethat all anglemeasuresare congruent(note,this is NOTtrue for
otherpolygons.For example,both squaresand “long”rectangleshaveall angles,but they are not
similar).Use logic,and the informationpresentedaboveto completeExample1.
Example 1
Justifythe statementthat.