+0

# 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

0
1123
2
+1760

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

#1
+26329
+15

My interpretation is as follows:

Alan  Nov 15, 2015
Sort:

#1
+26329
+15

My interpretation is as follows:

Alan  Nov 15, 2015
#2
+78754
+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

### 9 Online Users

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details