SOLUTION: Square ABCD has 8-inch sides, M is the midpoint of BC, and N is the intersection of AM and diagonal BD. Find the lengths of NB, NM, NA, and ND.

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Question 1154271: Square ABCD has 8-inch sides, M is the midpoint of BC, and N is the intersection of AM and diagonal BD. Find the lengths of NB, NM, NA, and ND.
Answer by Edwin McCravy(20059) About Me  (Show Source):
You can put this solution on YOUR website!



Draw PQ through N parallel to AB

 

AP%2F%28MQ%29=PD%2F%28QB%29

AD=AP%2BPD=8     MB=MQ%2BQB=4
   PD=8-AP        QB=4-MQ

AP%2F%28MQ%29=%288-AP%29%2F%284-MQ%29

AP%284-MQ%29=MQ%288-AP%29

4%2AAP-AP%2AMQ=8%2AMQ-MQ%2AAP

4%2AAP-AP%2AMQ=8%2AMQ-MQ%2AAP

4%2AAP-AP%2AMQ=8%2AMQ-AP%2AMQ

4%2AAP=8%2AMQ

AP=2%2AMQ

By similar triangles,

NA=2%2ANM

By the Pythagorean theorem,

AM%5E2=AB%5E2%2BBM%5E2
AM%5E2=8%5E2%2B4%5E2
AM%5E2=64%2B16
AM%5E2=80
AM=sqrt%2880%29
AM=sqrt%2816%2A5%29
AM=4sqrt%285%29
NA%2BNM=4sqrt%285%29
2%2ANM%2BNM=4sqrt%285%29
3%2ANM=4sqrt%285%29
NM=4sqrt%285%29%2F3
NA=2%2ANM=2%2A4sqrt%285%29%2F3=8sqrt%285%29%2F3

You finish by finding NB and ND the same way.

Edwin