My teacher says " the pythagorean theorem is the distance formula slimmed down. Is this true?

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Yes, the ditance formula is derived from the pythagorean theorem. Pyth Theorem is a^2 + b^2 = c^2. In the distance formula c is the distance you are looking for, a is (x2 - x1) and b is (y2-y1). So after squaring a, b and taking the square root of the expression, you get the actual distance formula.    

Yes this is true! Draw a right triangle. Draw a boat at the top of the triangle. Now draw a stick person (which is you) at the bottom right corner of the triangle. Now imagine you're in the sea and you need to swin to the boat for safety. Would it be faster to swim to the the bottom left corner triangle and then swim up to the boat or would it be faster to swim directly across, or the hypotenuse? I'm going to let you ask for yourself?