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# Help?

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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

May 29, 2020

#1
+21953
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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  =  AB

340 + 84  =  AB2

424  =  AB2

AB  =  sqrt(424)

May 29, 2020
#2
+188
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Wow! Thank you!

Noori  May 30, 2020