+297 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?
+129847 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
