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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?

Jun 11, 2020

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can you fix your LaTeX code? it is slightly hard to read.

Jun 11, 2020
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In right triangle ABC angle C is 90 degrees. Median AM has a length of 19 and median BN has a length of 13. What is the hypotenuse of the traingle.

Jun 11, 2020
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Let BC = a, AC = b.

Now, $$\overline{AM} = \sqrt{b^2 + \left(\dfrac a2\right)^2} = \dfrac12 \sqrt{4b^2 + a^2}$$

and $$\overline{BN} = \sqrt{a^2+ \left(\dfrac{b}{2}\right)^2} = \dfrac12 \sqrt{4a^2 + b^2}$$

According to the question, AM = 19 and BN = 13.

$$4b^2 + a^2 = 38^2 \\4a^2 + b^2 = 26^2$$

$$a^2 + b^2 = \dfrac{38^2 + 26^2}{5}$$
The length of the hypotenuse is $$\sqrt{a^2+ b^2} = 2\sqrt{106}$$