How would you set up the distance formula to find the distance between a and b?

(The coordinates for the line segment are ( A (-3,4) and B(2,-3))
A. d = (square root) (-4-3)^2 + (-3-2)^2
B. d = (square root) (2--3)^2+(-3-4)^2
C. d = (square root) (-3-4)^2 + (2--3)^2
D. d = (square root) (-4+3^2+(3--2)^2

1 answer

To find the distance \(d\) between two points \(A(x_1, y_1)\) and \(B(x_2, y_2)\), you can use the distance formula:

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

In this case, the coordinates of the points \(A\) and \(B\) are given as \(A(-3, 4)\) and \(B(2, -3)\).

Let's identify the coordinates:

  • \(x_1 = -3\), \(y_1 = 4\) (from point \(A\))
  • \(x_2 = 2\), \(y_2 = -3\) (from point \(B\))

Now, applying the distance formula:

\[ d = \sqrt{(2 - (-3))^2 + (-3 - 4)^2} \] \[ = \sqrt{(2 + 3)^2 + (-3 - 4)^2} \] \[ = \sqrt{(5)^2 + (-7)^2} \] \[ = \sqrt{25 + 49} \] \[ = \sqrt{74} \]

Now, looking at the options you provided, let's find the one that correctly applies the distance formula:

A. \(d = \sqrt{(-4-3)^2 + (-3-2)^2}\) (incorrect)

B. \(d = \sqrt{(2 - (-3))^2 + (-3 - 4)^2}\) (correct)

C. \(d = \sqrt{(-3 - 4)^2 + (2 - (-3))^2}\) (incorrect)

D. \(d = \sqrt{(-4 + 3)^2 + (3 - (-2))^2}\) (incorrect)

Thus, the correct answer is B: \(d = \sqrt{(2 - (-3))^2 + (-3 - 4)^2}\).