To find the length of the third side of a right triangle when the lengths of the other two sides are given, you can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.
Given:
- One side (leg) = 6
- The other side (leg) = 10
- The hypotenuse = \( b \)
According to the Pythagorean theorem:
\[ b^2 = 6^2 + 10^2 \]
Calculating the squares:
\[ b^2 = 36 + 100 \] \[ b^2 = 136 \]
Now, to find \( b \):
\[ b = \sqrt{136} \]
Next, we simplify \( \sqrt{136} \):
\[ \sqrt{136} = \sqrt{4 \times 34} = \sqrt{4} \times \sqrt{34} = 2\sqrt{34} \]
So, the length of the hypotenuse (which is the third side) is:
\[ b = 2\sqrt{34} \]
Thus, the length of the third side is
\[ \boxed{2\sqrt{34}} \]