Question

in the formula ( d = sqrt { ( x _ { 2 } - x _ { 1 } ) ^ { 2 } + ( y _ { 2 } - y _ { 1 } ) ^ { 2 } } ), how does each subtraction expression relate to the pythagorean theorem?

each subtraction expression represents half the length of the hypotenuse of a right triangle with a hypotenuse of length ( d ).

each subtraction expression represents the length of the hypotenuse of a right triangle with a leg of length ( d ).

each subtraction expression represents the length of one leg of a right triangle with a hypotenuse of length ( d ).

each subtraction expression represents the square of the length of the hypotenuse of a right triangle with a hypotenuse of length ( d ).

Explanation:

The distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) is derived from the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\) (where \(c\) is the hypotenuse of a right - triangle and \(a,b\) are the legs). In the coordinate plane, if we have two points \((x_1,y_1)\) and \((x_2,y_2)\), the horizontal distance between them is \(|x_2 - x_1|\) (which can be considered as one leg of a right - triangle) and the vertical distance is \(|y_2 - y_1|\) (the other leg). The distance \(d\) between the two points is the length of the hypotenuse of the right - triangle formed by these horizontal and vertical segments. The subtraction expressions \((x_2 - x_1)\) and \((y_2 - y_1)\) give the lengths of the legs of the right - triangle.

Answer:

Each subtraction expression represents the length of one leg of a right triangle with a hypotenuse of length \(d\).