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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?

 Apr 30, 2024
 #1
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A

 

N

 

C      M       B

 

AC^2  +  (BC/2)^2  = 1

(AC/2)^2 + BC^2 = 1           simplify this system

 

AC^2 + BC^2 / 4  =  1

AC^2/4 + BC^2 = 1

 

4AC^2 + BC^2  = 4

-AC^2/4 - BC^2  =  -1       add these

 

(15/4)AC^2  =  3

AC^2  = 4/5

 

 

And

 

AC^2 + (BC/2)^2  = 1

4/5 + BC^2 / 4  =  1

BC^2 / 4  = 1/5

BC^2  = 4/5

 

 

Hypotenuse length =  sqrt [ AC^2 + BC^2 ] =   sqrt [ 4/5 + 4/5 ] =  sqrt (8/5) = 2sqrt (2) / sqrt (5)  =

 

2 sqrt (10) /  5

 

cool cool cool

 Apr 30, 2024

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