The length of the median to the hypotenuse of an isosceles, right triangle is $10$ units.  What is the length of a leg of the triangle, in units? Express your answer in simplest radical form.
I think we can use the Pythagorean theorem to solve this problem. Right. The Pythagorean theorem is $a^2+b^2 = c^2$. Right. Now let's call the median to the hypotenuse $m$. We are trying to find $a$, correct? That is correct. Now we just need to find a relationship between $a$, $m$, and $c$. Do you remember the formula for the length of a median? Vaguely. But let's derive it together. A median is a line segment drawn from one vertex of a triangle to the midpoint of the opposite side. Yes. And we can use the Pythagorean theorem to find the length of that line segment. Ok. So in this case we have a right triangle with hypotenuse $c$ and one of the legs is $a$. The midpoint of the hypotenuse would be $\frac{c}{2}$. We know that $c=\sqrt{a^2+b^2}$. But since $a=b$, we can say that $c=\sqrt{2a^2}=a\sqrt{2}$. Yes. And if we draw the median, we form another right triangle. I see. So $m^2 + \left(\frac{c}{2}\right)^2 = a^2$. That's true. So now we have $m^2 + m^2 = a^2$. Which results in $m^2 + m^2 = a^2$. And since $m=10$, we have $10^2 + 10^2 = a^2$. Simplifying gives $2\cdot 10^2 = a^2$. So $a= 10\sqrt{2}$. And that is our final answer.

