Starting from point $$ A $$, a boat sails due south for 6 miles, then due east for 3 miles, then due south again for 4 miles. How far is the boat from point $$ A $$ ? The boat is (Simplify your answer. Type an exact answer, using radicals as needed.)

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1. Let the starting point $$ A $$ have coordinates $$ (0, 0) $$.

2. After sailing 6 miles due south, the boat reaches point $$ B $$ with coordinates:
   $$
   (0, 0 - 6) = (0, -6)
   $$

3. Next, the boat sails 3 miles due east to reach point $$ C $$ with coordinates:
   $$
   (0 + 3, -6) = (3, -6)
   $$

4. Finally, the boat sails 4 miles due south to reach point $$ D $$ with coordinates:
   $$
   (3, -6 - 4) = (3, -10)
   $$

5. The distance between point $$ A $$ and point $$ D $$ is given by the distance formula:
   $$
   d = \sqrt{(3 - 0)^2 + (-10 - 0)^2} = \sqrt{3^2 + (-10)^2} = \sqrt{9 + 100} = \sqrt{109}
   $$

The boat is $$\sqrt{109}$$ miles from point $$ A $$.
The boat is $$\sqrt{109}$$ miles from point $$ A $$.

Starting from point , a boat sails due south for 6 miles, then due east for 3 miles, then due south again for 4 miles. How far is the boat from point ?
The boat is
(Simplify your answer. Type an exact answer, using radicals as needed.)

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Starting from point , a boat sails due south for 6 miles, then due east for 3 miles, then due south again for 4 miles. How far is the boat from point ? The boat is (Simplify your answer. Type an exact answer, using radicals as needed.)