<s>[INST] Provide me an example about how to use the pythagorean theorem [/INST] The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. It can be written as an equation: a^2 + b^2 = c^2, where "c" is the length of the hypotenuse and "a" and "b" are the lengths of the other two sides.

Example:

Suppose you have a right triangle where one side has length 6 units and the other side has length 8 units. To find the length of the hypotenuse, you would use the Pythagorean theorem:

6^2 + 8^2 = c^2
36 + 64 = c^2
100 = c^2
c = 10

So, in this case, the length of the hypotenuse is 10 units.</s>\ <s>[INST] How is the Pythagorean theorem used to calculate the distance between two points? [/INST] In the same way. You find the horizontal distance and square that and the vertical distance and square that. And then you take the squareroot of the sum.

E.g. the points (4,5) and (7,1). The horizontal distance is 7-4=3, 3^2=9. And the vertical distance is 1 - 5 = -4. (-4)²=16. And the sum is 9+16=25. The square root of 25 is 5. So the distance between (4,5) and (7,1) is 5.</s>