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# Geometry

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In right triangle $ABC,$ $\angle C = 90^\circ$. Median $\overline{AM}$ has a length of 1, and median $\overline{BN}$ has a length of 1. What is the length of the hypotenuse of the triangle?

Jul 1, 2024

#1
+1271
+1

We can use a handy equation and the fact that it's a right triangle to solve this problem.

First, let's note some equations. Since AM and BN is the median, let's set a and b to be the two legs of the triangle. we have the equation

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

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

Now, these two equations are quite important, since we want to know what a^2+b^2 is. Plugging in 1 and 1, and isolating all terms except for a and b, we find that

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

Now, adding the two equations, we have the formula

$$5a^2+5b^2 = 8\\ a^2+b^2 = 8/5$$

Thus, the hypotenuse, which is equal to the square root of a^2 + b^2 is just $$\sqrt{8/5}$$

So our answer is just $$\sqrt{8/5}$$

Thanks! :)

Jul 1, 2024

#1
+1271
+1

We can use a handy equation and the fact that it's a right triangle to solve this problem.

First, let's note some equations. Since AM and BN is the median, let's set a and b to be the two legs of the triangle. we have the equation

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

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

Now, these two equations are quite important, since we want to know what a^2+b^2 is. Plugging in 1 and 1, and isolating all terms except for a and b, we find that

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

Now, adding the two equations, we have the formula

$$5a^2+5b^2 = 8\\ a^2+b^2 = 8/5$$

Thus, the hypotenuse, which is equal to the square root of a^2 + b^2 is just $$\sqrt{8/5}$$

So our answer is just $$\sqrt{8/5}$$

Thanks! :)

NotThatSmart Jul 1, 2024