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 the length of one leg 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 half the length of the hypotenuse 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:

Relate the distance formula to the Pythagorean theorem

$$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \implies d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 $$

Compare with the Pythagorean theorem formula

$$ c^2 = a^2 + b^2 $$

$$ \text{Let } c = d, \quad a = |x_2 - x_1|, \quad b = |y_2 - y_1| $$

Identify the geometric meaning of the subtraction expressions

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Answer:

• (A) Each subtraction expression represents the length of one leg of a right triangle with a hypotenuse of length d. (Correct answer)
• (B) Each subtraction expression represents the length of the hypotenuse of a right triangle with a leg of length d.
• (C) Each subtraction expression represents half the length of the hypotenuse of a right triangle with a hypotenuse of length d.
• (D) Each subtraction expression represents the square of the length of the hypotenuse of a right triangle with a hypotenuse of length d.