+219 In right triangle ABC, angle C is 90 degrees. Median AM has a length on 19, and the median BN has a length of 13. What is the length of the hypotenuse of the triangle? Triangle (ABC).
Thank you!!
Noori
+23246 Right triangle(ABC) with C the right angle.
M is the midpoint of BC.
N is the midpoint of AC.
Let AC = 2x, making CN = x.
Let CB = 2y, making CM = y.
Consider right triangle(ACM): Consider right triangle(BNC)"
CM2 + CA2 = AM2 BC2 + CN2 = BN2
y2 + (2x)2 = 192 (2y)2 + x2 = 132
y2 + 4x2 = 361 4y2 + x2 = 169
y2 + 4x2 = 361 ---> multiply by 4 ---> 4y2 + 16x2 = 1444
4y2 + x2 = 169 ---> multiply by -1 ---> -4y2 - x2 = -169
15x2 = 1275
x2 = 85
x = sqrt(85) ---> AC = 2·sqrt(85)
y2 + 4x2 = 361 ---> multiply by -1 ---> y2 - 4x2 = -361
4y2 + x2 = 169 ---> multiply by 4 ---> 16y2 + 4x2 = 676
15y2 = 315
y2 = 21
y = sqrt(21) ---> CB = 2·sqrt(21)
To find AB: AC2 + CB2 = AB2 ---> [ 2·sqrt(85) ]2 + [ 2·sqrt(21) ]2 = AB2
340 + 84 = AB2
424 = AB2
AB = sqrt(424)
