The lengths of the two legs of a right triangle are the two distinct roots of the quadratic $x^2 - 36x + 70$. What is the length of the triangle’s hypotenuse?
1. **Identify the roots using Vieta's formulas:**
   Given the quadratic equation \(x^2 - 36x + 70 = 0\), by Vieta's formulas, the sum of the roots \(x_1\) and \(x_2\) is given by:
   \[
   x_1 + x_2 = 36
   \]
   and the product of the roots is:
   \[
   x_1 x_2 = 70
   \]

2. **Apply the Pythagorean Theorem:**
   The length of the hypotenuse \(c\) of a right triangle with legs \(x_1\) and \(x_2\) is given by:
   \[
   c = \sqrt{x_1^2 + x_2^2}
   \]

3. **Express \(x_1^2 + x_2^2\) in terms of \(x_1 + x_2\) and \(x_1 x_2\):**
   Using the identity \((x_1 + x_2)^2 = x_1^2 + x_2^2 + 2x_1 x_2\), we can solve for \(x_1^2 + x_2^2\):
   \[
   (x_1 + x_2)^2 = x_1^2 + x_2^2 + 2x_1 x_2
   \]
   \[
   36^2 = x_1^2 + x_2^2 + 2 \cdot 70
   \]
   \[
   1296 = x_1^2 + x_2^2 + 140
   \]
   \[
   x_1^2 + x_2^2 = 1296 - 140
   \]
   \[
   x_1^2 + x_2^2 = 1156
   \]

4. **Calculate the hypotenuse:**
   \[
   c = \sqrt{x_1^2 + x_2^2} = \sqrt{1156} = 34
   \]

The final answer is \(\boxed{34}\).