In right triangle $ABC,$ $\angle C = 90^\circ.$ Median $\overline{AM}$ has a length of $19,$ and median $\overline{BN}$ has a length of $13.$ What is the length of the hypotenuse of the triangle?

  AB = 2√106 ≈ 20.591

Step-by-step explanation:

The Pythagorean theorem says the square of the hypotenuse is equal to the sum of the squares of the legs.

For median AM, we have ...

  AM² = CM² +AC² = (BC/2)² +AC²

For median BN, we have ...

  BN² = CN² +BC² = (AC/2)² +BC²

The sum of these two equations is ...

  AM² +BN² = BC²/4 +AC² +AC²/4 +BC² = (5/4)(AC² +BC²)

  AM² +BN² = (5/4)(AB²)

The hypotenuse of triangle ABC is then ...

  AB = √(4/5(AM² +BN²))

  AB = 2√((19² +13²)/5)

  AB = 2√106 ≈ 20.591