### Instruction:
Complex numbers $a,$ $b,$ and $c$ are zeros of a polynomial $P(z) = z^3 + qz + r,$ and $|a|^2 + |b|^2 + |c|^2 = 250.$ The points corresponding to $a,$ $b,$ and $c$ in the complex plane are the vertices of a right triangle with hypotenuse $h.$ Find $h^2.$

### Response:

What is the relationship between the roots of a polynomial and its coefficients? The sum of the roots of a polynomial is the negation of the next-to-leading coefficient divided by the leading coefficient. That's correct. Since the leading coefficient is 1, the sum of the roots is equal to the negative of the second coefficient. That's correct. Since $a,b,c$ are zeroes of the polynomial, we can say that $a+b+c = 0$ That's correct. The centroid is the point $(\frac{a_1+b_1+c_1}{3},\frac{a_2+b_2+c_2}{3})$, where $a_1,a_2,b_1,b_2,c_1,c_2$ are the coordinates of $a,b,c$ Yes, so we know that $\frac{a_1+b_1+c_1}{3} = 0$, and similarly for the second coordinate. That's correct. Now, what's the relationship between the coordinates of $a,b,c$ and the magnitudes $|a|^2,|b|^2,|c|^2$? The square of the magnitude of a complex number is the sum of the squares of its coordinates. That's right. $|a|^2 = a_1^2+a_2^2$, and similarly for $b$ and $c$. We're given that $|a|^2 + |b|^2 + |c|^2 = 250$. And we know that the centroid of a triangle is the arithmetic mean of the coordinates of the vertices of the triangle. But we know that the centroid of this triangle is the origin. Thus $a^2 + b^2 + c^2$ is the sum of the squares of the lengths of the segments from the origin to each vertex. Thus $a^2 + b^2 + c^2$ is equal to the sum of the squares of $\frac{2}{3}$ of the lengths of the medians of the triangle. But we know that the sum of the squares of the lengths of the medians of a triangle is equal to $\frac{3}{4}$ of the sum of the squares of the lengths of the sides of the triangle. $= \frac{4}{9} \cdot (length\ of\ median\ from\ a)^2 + \frac{4}{9} \cdot (length\ of\ median\ from\ b)^2 + \frac{4}{9} \cdot (length\ of\ median\ from\ c)^2$ But we know that the sum of the squares of the lengths of the medians of a triangle is equal to $\frac{3}{4}$ of the sum of the squares of the lengths of the sides of the triangle. Thus $|a|^2 + |b|^2 + |c|^2 = \frac{4}{9} \cdot \left(\frac{3}{4}\cdot(\text{sum of squares of lengths of sides of triangle})\right)$ And we're given that $|a|^2 + |b|^2 + |c|^2 = 250$. From this we can solve for the sum of the squares of the lengths of the sides of the triangle. That's right. $250 = \frac{4}{9} \cdot \left(\frac{3}{4}\cdot(\text{sum of squares of lengths of sides of triangle})\right)$. So we have that $\text{sum of squares of lengths of sides of triangle} = \frac{9}{4} \cdot \frac{4}{3} \cdot 250$. Right. So the sum of the squares of the lengths of the sides of the triangle is $250 \cdot 3 = 750$. And we know that in a right triangle with legs $a,b$ and hypotenuse $h$, $a^2 + b^2 = h^2$ From this we can solve for $h^2$. So if $a^2 + b^2 + h^2 = 750$ then $h^2 + h^2 = 750$, and so $2h^2 = 750$. So $h^2 = \frac{750}{2}$ And so $h^2 = 375$ And that's our final answer.

