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avatar+1211 

In right triangle $ABC,$ $\angle C = 90^\circ$. Median $\overline{AM}$ has a length of 1, and median $\overline{BN}$ has a length of 1. What is the length of the hypotenuse of the triangle?

 Mar 20, 2024
 #1
avatar+128794 
+1

A

 

N       

 

C        M           B

 

AC^2  + [ (BC)/2 ] ^2  = 1

(AC/2)^2  + BC^2  =  1

 

AC^2  + BC^2/4  =  1

AC^2/4  + BC^2  = 1

 

4AC^2 + BC^2  =  4

AC^2/4  +  BC^2  =  1          subtract these

 

(15/4)AC^2  =  3

AC^2  = 12/15  = 4/5

 

 

(12/15)/2 + BC^2 =  1

(2/5) + BC^2  =1

BC^2  = 4/5

 

AB =  sqrt  [ AC^2 + BC^2 ]   =  sqrt [  4/5 + 4/5 ]  = sqrt [ 8/5]  =  (2/5)sqrt (10) 

 

cool cool cool

 Mar 20, 2024

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