A median of a triangle is a line segment from a vertex of a triangle to the midpoint of the opposite side of the triangle. The medians to the legs of a certain right triangle have lengths 13 and 19. What is the length of the hypotenuse of the triangle?

Mellie
Nov 14, 2015

#2**+10 **

Let L1 be the length of one of the legs and L2 be the lentgh of the other.......and by the Pythagorean Theorem ,we have that

[L1]^2 + [(1/2)L2] ^2 = 19^2 and

[L2]^2 + [(1/2)L1]^2 = 13^2 simplify

L1^2 + (1/4)L2^2 = 361 (1)

L2^2 + (1/4)L1^2 = 169 (2)

Rearranging (1),we have

L1^2 = 361 - (1/4)L2^2 (3) and subbing this into (2),we have

L2^2 +(1/4)[ 361 - (1/4)L2^2 ] = 169 multiply through by 16

16L2^2 + 4*361 - L2^2 = 2704

15L2^2 = 2704 - 4*361

15L2^2 = 1260 divide by 15

L2^2 = 84 → L2 = 2sqrt(21)

And using (3), we have

L1^2 = 361 - (1/4)L2^2

L1^2 = 361 - (1/4)84

L1^2 = 361 - 21

L1^2 = 340 → L1 = 2sqrt(85)

So....the hypotenuse = sqrt (84 + 340) = sqrt (424) = 2sqrt (106)

CPhill
Nov 15, 2015