CK12 - Geometry

In the figureabove,the big triangle is a right trianglewith right angle and

and. So, if , , and are similar, they will all haveanglesof

, , and.

First look at. , and. Sincethe sum of the threeanglesin a triangle

alwaysequals , the missingangle, , mustmeasure , since.

Liningup the congruentangles,we can write.

Now look at. has a measureof , and. Sincethe sum of the three

anglesin a trianglealwaysequals , the missingangle, , mustmeasure , since

Now, sincethe triangleshavecongruentcorresponding angles, and

are similar.

Thus,. Theiranglesare congruentand their sidesare proportional.

Notethat you mustbe very carefulto matchup correspondingangleswhenwritingtrianglesimilaritystate-

ments.Herewe shouldwrite. This exampleis challengingbecausethe
trianglesare overlapping.

GeometricMeans

Whensomeoneasksyou to find the averageof two numbers,you probablythinkof the arithmeticmean
(average).Chancesare goodyou’veworkedwith arithmeticmeansfor manyyears,but the conceptof a
geometricmeanmay be new. An arithmeticmeanis foundby dividingthe sum of a set of numbersby the
numberof itemsin the set. Arithmeticmeansare usedto calculateoverallgradesand manyotherapplications.
The big idea behindthe arithmeticmeanis to find a “measureof center”for a groupof numbers.

A geometricmeanappliesthe sameprinciples,but relatesspecificallyto size,length,or measure.For ex-
ample,you may havetwo line segmentsas shownbelow. Insteadof addingand dividing,you find a geometric
meanby multiplyingthe two numbers,then findingthe squareroot of the product.

To find the geometricmeanof thesetwo segments,multiplythe lengthsand find the squareroot of the
product.