Let's say I have a right triangle (ABC) with two medians drawn A to the side of BC, which cuts it in half, and the median from BC that cuts side AC in half. Then, how would I find the length of the hypotenuse? Thank you!

ant101 Jun 26, 2019

#1**0 **

Wait, I can label the bottom parts as 1/2a and 1/2a, and I can label the top parts as 1/2b and 1/2b.

There are also two triangles at the top, so I can use the Pythagorean Theorem?

ant101 Jun 26, 2019

#3**+1 **

Ant, I assume we know the length of the medians....if so....

Call one median AD and the other BE

Then,,,,we have two right triangles ADC and BEC

So.....this sets up a system of equations using the Pythagorean Theorem

[ (1/2)AC]^2 + [BC]^2 = [BE]^2 ⇒ (1/4)AC^2 + BC^2 = BE^2 (1)

[ (AC]^2 + [ [(1/2)BC]^2 = [ AD]^2 ⇒ AC^2 + (1/4)BC^2 = AD^2 (2)

Multiply (1) by -4 and we have

-AC^2 - 4BC^2 = - 4BE^2

AC^2 + (1/4)BC^2 = AD^2 add these

(-15/4)BC^2 = -4BE^2 + AD^2

Since we assume that we know BE and AD, we can solve this for BC and then use either (1) or (2) to find AC

Then.....just use thePythagorean Theorem again to find AB

CPhill Jun 27, 2019