Geometry: An Interactive Journey to Mastery



Lesson 6: Distance, Midpoints, and Folding Ties

Example 1

Let A ͼ௘௘ͽB ͼ௘௘ͽDQG5 ͼ௘௘ͽ

D௘ͽ 6KRZWKDWAR = RB.

E௘ͽ 6XSSRVHWKDWS ͼ௘k௘ͽLVDQRWKHUSRLQWHTXLGLVWDQWIURPA and Bͼ௘WKDWLVVDWLV¿HVAS = SB௘ͽ:KDWLV

WKHYDOXHRIk"

Solution

D௘ͽ :HKDYHAR ^22 and BR ^22 7KHVHDUHWKHVDPHYDOXH

E௘ͽ :HQHHG ^22   kk^22

6TXDULQJHDFKVLGHDQGIROORZLQJWKURXJKZLWKWKHDOJHEUDJLYHVWKHIROORZLQJ

 22  ^22 

     

    



kk

kk k k

kkk

k

 

  

 

Example 2

)LQGWKHPLGSRLQWMRIWKHOLQHVHJPHQWFRQQHFWLQJA ͼ௘í௘ͽWRB ͼ௘௘ͽ

6KRZWKDWAMHTXDOVMBWRYHULI\WKDWMLVLQGHHGHTXLGLVWDQWIURPA and B.

$OVRYHULI\WKDWAMMBHTXDOVABWRVKRZWKDWM does indeed lie on AB.

Solution

7KHPLGSRLQWLVM §·§· ̈ ̧ ̈ ̧©¹©¹  22 22

7KHGLVWDQFHEHWZHHQ ̈ ̧§·©¹ 22  and   is

^2222       

AM   §·§·§·§· ̈ ̧ ̈ ̧ ̈ ̧ ̈ ̧©¹©¹©¹©¹