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