Answered

In right triangle $ABC,$ $\angle C = 90^\circ.$ Median $\overline{AM}$ has a length of $19,$ and median $\overline{BN}$ has a length of $18.$ What is the length of the hypotenuse of the triangle?



Answer :

The length of the hypotenuse of the triangle is; AB = 2√106 ≈ 20.591

How to use the Pythagorean theorem?

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

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 gives us;

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

AM² +BN² = (5/4)(AB²)

Thus, we conclude that the length of the hypotenuse of triangle ABC is;

AB = √(4/5(AM² +BN²))

AB = 2√((19² +13²)/5)

AB = 2√106 ≈ 20.591

Read more about Pythagorean theorem at; https://brainly.com/question/343682

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